Conjectures related to aliquot sequences starting on integer powers n^i

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Access the topic of publication of the conjectures : 139 conjectures in total.

Conjectures (1) to (133) published on August 19, 220 on the Mersenne forum, see post #447

In all the statements below, k is an integer.

Note: Several of these conjectures motivated my request to Edwin Hall to push the calculations further for some exponents i=36*k, i=60*k, i=70*k, i=72*k, i=90*k.

Conjecture (1)* :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(2*k).
s(2^(2*k)) = 2^(2*k) - 1 = M2k, which is divisible by M2 = 3.

Conjecture (2)** :

The prime number 3 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(4*k).
s(24k) = 24k - 1 = (22k - 1)(22k + 1).
22k + 1 is 2 (mod 3), so there exist a prime p such that p is 2 (mod 3) and p2m - 1 divide 22k + 1, but p2m not divide 22k + 1.
gcd(22k - 1, 22k + 1) = gcd(2, 22k + 1) = 1, so no other factor of p from 22k - 1.
Hence p2m-1 preserved 3 for s(24k), so 3 divide s(s(24k)).

See successively on the Mersenne forum, posts #476, #480 and #481

Conjecture (3) :

The prime number 3 appears in the decomposition of the terms of indexes 1 through 7 of all sequences that begin with the integers 2^(36*k).

Conjecture (4) :

The prime number 3 appears in the decomposition of the terms of indexes 1 through 18 of all sequences that begin with the integers 2^(126*k).

Conjecture (5)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(4*k).
s(2^(4*k)) = 2^(4*k) - 1 = M4k, which is divisible by M4 = 15 = 3*5.

Conjecture (6)** :

The prime number 5 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(28*k), 2^(44*k), 2^(76*k), 2^(92*k), 2^(116*k).

Conjecture (7) :

The prime number 5 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(36*k).

Conjecture (8) :

The prime number 5 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(132*k).

Conjecture (9)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(3*k).
s(2^(3*k)) = 2^(3*k) - 1 = M3k, which is divisible by M3 = 7.

Conjecture (10)** :

The prime number 7 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(12*k).

Conjecture (11) :

The prime number 7 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(60*k).

Conjecture (12)* :

The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(10*k).

Conjecture (13)** :

The prime number 11 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(120*k), 2^(130*k).

Conjecture (14) :

The prime number 11 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(70*k).

Conjecture (15) :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(12*k).

Conjecture (16)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(60*k).

Conjecture (17)* :

The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(8*k).

Conjecture (18)** :

The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(144*k).

Conjecture (19) :

The prime number 19 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(72*k).

Conjecture (20) :

The prime number 31 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(90*k).
Conjecture invalidated by Edwin Hall's calculations.

Conjecture (21)** :

The prime number 79 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(156*k).

Conjecture (22)** :

The prime number 2089 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(87*k).

Conjecture (23)** :

The prime number 4051 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(100*k).

Conjecture (24) :

The prime number 15121 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(540*k).

Conjecture (25)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 3^(4*k).

Conjecture (26)** :

The prime number 5 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(4+8*k).

Conjecture (27)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(6*k).

Conjecture (28)* :

The prime number 7 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 3^(6+12*k).
Here is the observation that led to this conjecture:
Code:
prime 7 in sequence 3^6 at index i for i from 1 to 5
prime 7 in sequence 3^18 at index i for i from 1 to 50
prime 7 in sequence 3^30 at index i for i from 1 to 25
prime 7 in sequence 3^42 at index i for i from 1 to 86
prime 7 in sequence 3^54 at index i for i from 1 to 179
prime 7 in sequence 3^66 at index i for i from 1 to 39
prime 7 in sequence 3^78 at index i for i from 1 to 124
prime 7 in sequence 3^90 at index i for i from 1 to 171
prime 7 in sequence 3^102 at index i for i from 1 to 72
prime 7 in sequence 3^114 at index i for i from 1 to 45
prime 7 in sequence 3^126 at index i for i from 1 to 60
prime 7 in sequence 3^138 at index i for i from 1 to 230
prime 7 in sequence 3^150 at index i for i from 1 to 148
prime 7 in sequence 3^162 at index i for i from 1 to 228
prime 7 in sequence 3^174 at index i for i from 1 to 219
prime 7 in sequence 3^186 at index i for i from 1 to 9
prime 7 in sequence 3^198 at index i for i from 1 to 105
prime 7 in sequence 3^210 at index i for i from 1 to 194
prime 7 in sequence 3^222 at index i for i from 1 to 98
prime 7 in sequence 3^234 at index i for i from 1 to 87
prime 7 in sequence 3^246 at index i for i from 1 to 38
But on reflection, this conjecture is not extraordinary.
7 is a prime number which is in the dcomposition of the 2^2*7 driver.
It is therefore normal that it persists in so many consecutive terms.
On the other hand, it should be shown here that s(3^(6+12*k)) has the driver 2^2*7 as a factor.

Conjecture (29)* :

The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(5*k).

Conjecture (30)* :

The prime number 13 appears in the decomposition of index 1 terms in all sequences that begin with the integers 3^(3*k).

Conjecture (31)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(51*k).

Conjecture (32)* :

The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(16*k).

Conjecture (33)** :

The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(48*k).

Conjecture (34) (* if only index 1) :

The prime number 19 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 3^(18*k).

Conjecture (35) (* if only index 1 and 2):

The prime number 19 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(36*k).

Conjecture (36)* :

The prime number 23 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(11*k).

Conjecture (37)** :

The prime number 31 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(30*k).

Conjecture (38)* :

The prime number 37 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(18*k).

Conjecture (39)** :

The prime number 37 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(36*k).

Conjecture (40)** :

The prime number 79 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(78*k).

Conjecture (41)** :

The prime number 547 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 3^(14*k).

Conjecture (42)**, already known conjecture, see previous posts :

The prime number 398581 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(26*k).

Conjecture (43) :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 5^(2*k).

Conjecture (44) :

The prime number 3 appears in the decomposition of many consecutive indexes of all sequences that begin with the integers 5^(2+4*k).

For example, 3 appears in the decomposition of the terms in indexes 1 through 786 of the sequence that begins with 5^58.

Conjecture (45)* :

The first number 5 never appears in the decomposition of the terms at index 1 of all sequences beginning with the integers 5^(k).

Conjecture (46)* :

The prime number 7 appears in the decomposition of terms at index 1 of all sequences that begin with the integers 5^(6*k).

Conjecture (47)** :

The prime number 7 appears in the decomposition of index 1 and index 2 terms of all sequences that begin with the integers 5^(12*k).

Conjecture (48)* :

The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(5*k).

Conjecture (49)** :

The prime number 11 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(35*k).

Conjecture (50)** :

The prime number 11 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 5^(40*k).

Conjecture (51)** :

The prime number 11 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 5^(65*k).

Conjecture (52)* :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(4*k).

Conjecture (53)* :

The prime number 17 appears in the decomposition of index 1 terms in all sequences that begin with the integers 5^(16*k).

Conjecture (54)* :

The prime number 19 appears in the decomposition of index 1 terms in all sequences that begin with the integers 5^(9*k).

Conjecture (55)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(18*k).

Conjecture (56)* :

The prime number 31 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(3*k).

Conjecture (57)** :

The prime number 31 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(30*k).

Conjecture (58) :

The prime number 31 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 5^(48+96*k).
Here is the observation that led to this conjecture:
Code:
prime 31 in sequence 5^48 at index i for i from 1 to 447
prime 31 in sequence 5^144 at index i for i from 1 to 32
The same remark can be made here as for the conjecture (28).
And it should be shown here that s(5^(48+96*k)) has the driver 2^4*31 as a factor.

Conjecture (59)* :

The prime number 71 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(5*k).

Conjecture (60)** :

The prime number 71 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(45*k).

Conjecture (61)* :

The prime number 521 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(10*k).

Conjecture (62)** :

The prime number 521 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(50*k).

Conjecture (63) :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that start with the integers 6^(1+2*k).

Conjecture (64)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(2*k).

Conjecture (65)* :

The prime number 7 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(6*k).

Conjecture (66)* :

The prime number 11 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(10*k) and 6^(2+10*k).

Conjecture (67)* :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(12*k).

Conjecture (68)* :

The prime number 19 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(18*k) and 6^(10 + 18*k).

Conjecture (69)* :

The prime number 23 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(11*k).

Conjecture (70)* :

The prime number 29 appears in the decomposition of index 1 terms in all sequences that begin with the integers 6^(28*k).

Conjecture (71)* :

The prime number 31 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(30*k), 6^(11+30*k) and 6^(17 + 30*k).

Conjecture (72)** :

The prime number 37 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 6^(36*k)

Conjecture (73)* :

The prime number 37 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 6^(14+36*k).

Conjecture (74)* :

The prime number 59 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(58*k), 6^(8+58*k), 6^(35+58*k) and 6^(53+58*k).

Conjecture (75)* :

The prime number 61 appears in the decomposition of the terms of index 1 of all sequences beginning with the integers 6^(60*k), 6^(44+60*k) and 6^(55+60*k).

Conjecture (76)* :

The prime number 71 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(70*k), 6^(11+70*k), 6^(32+70*k), 6^(35+70*k), 6^(46+70*k) and 6^(67+70*k).

Conjecture (77)* :

The prime number 601 appears in the decomposition of the terms of index 1 of all sequences beginning with the integers 6^(75*k).

Conjecture (78) :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 7^(3*k).

Conjecture (79) :

The prime number 3 appears in the decomposition of the terms from index 1 to 10 for all sequences that begin with the integers 5^(6+12*k) and 5^(21*k).

Conjecture (80)* :

The prime number 5 appears in the decomposition of the terms of index 1 for all sequences that begin with the integers 7^(4*k).

Conjecture (81)* :

The prime number 7 appears in the decomposition of index 1 terms in all sequences that begin with the integers 7^(3*k).

Conjecture (82)* :

The prime number 7 never appears in index 1 of all sequences that begin with the integers 7^(k).

Conjecture (83)* :

The prime number 11 appears in the decomposition of the terms in index 1 of all sequences that begin with the integers 7^(10*k).

Conjecture (84)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(12*k).

Conjecture (85) :

The prime number 13 appears in the decomposition of the terms of many indexes of all sequences that begin with the integers 7^(72*k).
27 consecutive indexes for 7^72 and 9 consecutive indexes for 7^144.

Conjecture (86)* :

The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(16*k).

Conjecture (87)** :

The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(32*k).

Conjecture (88)* :

The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(3*k).

Conjecture (89)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(9*k).

Conjecture (90)* :

The prime number 31 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(15*k).

Conjecture (91)** :

The prime number 67 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(66*k).

Conjecture (92)* :

The prime number 419 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(19*k).

Conjecture (93)** :

The prime number 419 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(38*k).
Checked up to k=23.

Conjecture (94) :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 10^(2*k).

Conjecture (95)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(3+4*k).

Conjecture (96)* :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(12*k).

Conjecture (97)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 10^(2+12*k).

Conjecture (98)** :

The prime number 19 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 10^(18*k).

Conjecture (99)* :

The prime number 61 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(52+60*k) and 10^(54+60*k).

Conjecture (100)** :

The prime number 61 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 10^(60*k).

Conjecture (101)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(3*k).

Conjecture (102)** :

The prime number 7 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 11^(6*k).

Conjecture (103)* :

The prime number 11 never appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(k).

Conjecture (104)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 11^(k).

Conjecture (105)* :

The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(3*k).

Conjecture (106)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 11^(6*k).

Conjecture (107)** :

The product of prime 19*79*547 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 11^(39*k).
REMARKABLE, checked up to k=12. See prime 79 bases 2 and 3.

Conjecture (108) :

The prime number 3 never appears in the decomposition of the terms of index 1 of all sequences that start with the integers 12^(k).

Conjecture (109)* :

The prime number 17 appears in the decomposition of index 1 terms in all sequences that begin with the integers 12^(16*k) and 12^(6+16*k).

It is difficult to notice for base 12, other behaviors different from the bases already presented so far.

Conjecture (110)* :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that start with the integers 13^(3*k).

Conjecture (111) :

The prime number 3 appears in the decomposition of the terms of indexes 1 through 6 of all sequences that begin with the integers 13^(6*k).

Conjecture (112)* :

The prime number 5 appears in the decomposition of the terms in index 1 of all sequences that begin with the integers 13^(4*k).

Conjecture (113) :

The prime number 5 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 13^(8+16*k).

Conjecture (114)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(2*k).

Conjecture (115) :

The prime number 7 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 13^(4+8*k).

Conjecture (116)* :

The prime number 13 never appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(k).

Conjecture (117)* :

The prime number 19 appears in the decomposition of index 1 terms in all sequences that begin with the integers 13^(18*k).

Conjecture (118)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(36*k).

Conjecture (119)* :

The prime number 29 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(14*k).

Conjecture (120)** :

The prime number 29 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(42*k).

Conjecture (121)* :

The prime number 61 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(3*k).

Conjecture (122)** :

The prime number 61 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(21*k).

Conjecture (123)** :

The prime number 3 appears in the decomposition of the terms of indexes 1 to 4 of all sequences starting with the integers 14^(6*k).

Conjecture (124)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 14^(4*k).

Conjecture (125)** :

The prime number 5 appears in the decomposition of the terms of indexes 1 to 4 of all sequences that begin with the integers 14^(1+4*k).

It is difficult to notice for base 14, other behaviors different from the bases already presented so far...

Conjecture (126)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences which start with the integers 15^(2*k), except for the 15^(8+12*k).

Conjecture (127)** :

The prime number 3 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(2*k).

Conjecture (128)** :

The prime number 5 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(4*k).

Conjecture (129)** :

The prime number 7 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(6*k).

Conjecture (130)** :

The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 17^(9*k).

Conjecture (131)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 17^(36*k).

Conjecture (132)** :

The prime number 229 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 17^(19*k).

Conjecture (133)** :

The prime number 229 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 17^(38*k).

Conjectures (134) to (137) published on March 2, 2021 on the Mersenne forum, see post #921

Conjecture (134) :

If a base b = p# is primorial (p prime > 7), then the sequence that starts with the integer b^14 is increasing from index 1 for a few iterations.

Note 1 : In general, this does not seem to be the case with the other exponents, except exponent 8 (see conjecture 135).
Note 2 : We believe that if we compute the following larger primorial bases, the growth phenomenon will occur with other exponents. To check.
Note 3 : This conjecture is completed and replaced by conjecture (140).

Conjecture (135) :

If a base b = p# is primorial (p prime > 29), then the sequence that starts with the integer b^8 is increasing from index 1 for a few iterations.

Note 1 : In general, this does not seem to be the case with the other exponents, except exponent 14 (see conjecture 134).
Note 2 : The same as for conjecture (134).
Note 3 : This conjecture is completed and replaced by conjecture (140).

Conjecture (136) :

If a base b = (p#) / 2 is primorial without the factor 2 (p prime> 3), then some sequences of this base grow from index 1 for a few iterations.

Note 1 : Until March 2021, we have only found three other odd bases for which this is also the case :
231 (3 * 7 * 11), 3003 (3 * 7 * 11 * 13) and 51051 (3 * 7 * 11 * 13 * 17),
to be seen as a primorial numbers without the factors 2 and 5 ?
Because 3003^5 and 51051^11 also have this property !
Note 2 : It is possible that this is just an illusion, maybe there are many other odd numbers that have the property ?
Note 3 : many of the exponents are prime numbers (especially 11 and 23) and not prime exponents are often equal to 7 * 5.

Conjecture (137) :

Base 2 sequences starting with 2^(12 * k), 2^(40 * k), 2^(90 * k), 2^(140 * k), 2^(210 * k), 2^(220 * k), 2^(330 * k), are increasing from index 1 for a few iterations.

Note 1 : This is not the case for the other exponents we have examined.
Note 2 : We believe that there must be other exponents of the form z * k (with z>330) which have this property.
Note 3 : We think that this phenomenon is related to the theorem which says that if p prime, s(p^i) is a factor of s(p^(i * m)) for every positive integer m, see post #466.
This theorem ensures, for example, that exponents multiple of 12 have many prime factors in their decomposition (like 2^12 itself), which ensures them growth for a few iterations.
Because of this mechanism, this conjecture looks more like those of post #447, see below.

Conjectures (138) to (139) published on March 6, 2021 on the Mersenne forum, see post #952

Conjecture (138) :

Base 6 sequences starting with 6^((2^3 * 3^2 * 5 * 7) * k) are increasing at least from index 1 to 2.
If we prove that:
2^(2^3 * 3^2 * 5 * 7 * k + 1) - 1 == 1 mod d
3^(2^3 * 3^2 * 5 * 7 * k + 1) - 1) / 2 == 1 mod d
6^(2^3 * 3^2 * 5 * 7 * k) == 1 mod d
Then s(6^(2^3 * 3^2 * 5 * 7 * k)) = (2^(2^3 * 3^2 * 5 * 7 * k + 1) - 1) * 3^(2^3 * 3^2 * 5 * 7 * k + 1) - 1) / 2 - 6^(2^3 * 3^2 * 5 * 7 * k) = 1 * 1 - 1 mod d
For p prime, and a such that gcd(a,p) = 1, we must have a^(p - 1) == 1 mod p.
So if p - 1 divides 2^3 * 3^2 * 5 * 7, it means 2^(2^3 * 3^2 * 5 * 7 * k) == 1^k == 1 mod p, hence 2^(2^3 * 3^2 * 5 * 7 * k + 1)-1 == 1^k * 2 - 1 == 1 mod p.
Same thing for 3 and 6.
That take care of p = 5+, 7+, 11, 13, 19, 29, 31, 37, 41, 43, 61, 71, 73, 127, 181, 211, 281, 421, 631
(+ This only prove that 5, 7 divide s(6^(2^3 * 3^2 * 5 * 7 * k)), not 5^2, 7^2)
For p = 337, we have (p - 1)/2 divides 2^3 * 3^2 * 5 * 7. Possible value for a^((p - 1)/2) is +-1 mod p. I guess that mean 2^((337 - 1)/2), 3^((337 - 1)/2),6^((337 - 1)/2) are happened to be 1 mod 337.
To prove that 5^2, 7^2 divide s(6^(2^3 * 3^2 * 5 * 7 * k)), we use the fact that if a == 1 mod p, then a^p == 1 mod p^2.
(5 - 1) divides 2^3 * 3^2 * 7, so a^(2^3 * 3^2 * 5 * 7) == 1 mod 5^2 for (a,p) = 1
(7 - 1) divides 2^3 * 3^2 * 5, so a^(2^3 * 3^2 * 5 * 7) == 1 mod 7^2 for (a,p) = 1

See successively on the Mersenne forum, posts #953, #954 and #960

Conjecture (139) :

There exists for each base b a starting exponent i, such that for any integer k, the sequences b^(i * k) are increasing from index 1 during at least one iteration.

Conjectures (140) published on April 7, 2021 on the Mersenne forum, see post #1074

Conjecture (140) :

If a base b = p# is primorial (p prime > 41), then s(b^(2 + 6 * k)) is abundant.
This new conjecture completes and replaces conjectures (134) and conjecture (135).