A343144 -id:A343144 - cp's OEIS frontend

A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.

3, 5, 17, 65, 257, 65537, 4294967297

Offset: 1

Jaroslav Krizek, Dec 16 2021

Corresponding pairs of values [tau(m-2), tau(m-1)]: [1, 2], [2, 3], [4, 5], [6, 7], [8, 9], [16, 17], [32, 33], ...
There are no other terms <= 2^1206 + 1 (from A046801 data).
The first 5 known Fermat primes from A019434 are in this sequence. Corresponding values of tau(A019434(n - 2)): 1, 2, 4, 8, 16, ...
Conjecture 1: Also numbers m of the form 2^k + 1 such that tau(m - 2) = k.
Conjecture 2: If 6th Fermat prime F_p6 exists, then tau(F_p6 - 2) is a power of 2 and tau(F_p6 - 1) = tau(F_p6 - 2) + 1.
Conjecture 3: Sequence is finite with 7 terms; supersequence of A262534.

			For number 257 holds: tau(255) = 8, tau(256) = 9.

Cf. A000005, A019434, A262534, A343144, A347078.

Intersection of (A055927+2) and A000051.

[2^k + 1: k in [1..50] | #Divisors(2^k) - #Divisors(2^k-1) eq 1];

A353032 a(n) is the smallest number m with n divisors such that m+1 has n-1 divisors, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 4, 8, 81, 0, 0, 0, 441, 6723, 0, 0, 0, 0, 767495140624, 2024, 665416609183179841, 0, 0, 0, 2050624, 263168, 0, 0, 670801950625, 0, 10871232294189453124, 532899, 0, 0, 0, 0, 67634176, 0, 55471075527984793933106579132930662929175947116953798971172816083061185149078369140624

Offset: 1

Jaroslav Krizek, Apr 18 2022

nonn

For n > 33, a(64) = 6890624 is the only positive term <= 10^8.
There is no number m <= 10^10 that is the first start of run of 3 consecutive integers m, m+1 and m+2 with triplet [tau(m), tau(m+1), tau(m+2)] = [tau(m), tau(m) - 1, tau(m) - 2].
If a(11) > 0 then a(11) > 10^100. - Charles R Greathouse IV, Apr 20 2022
a(36) = 1626347583, a(40) = 1173953168, a(49) = 304006671424, a(65) = 25221297570561, a(81) = 15579533124, a(96) = 68195356770303, a(100) = 1698353697680, a(136) = 28528257204224, a(256) = 334435516415. - Jon E. Schoenfield, Apr 24 2022
From Jon E. Schoenfield, May 01 2022: (Start)
a(35) is the smallest m such that m = 16*p^6 = q^16*r - 1 where p, q, and r are odd primes; a(35) <= 16*123024356097427^6 (an 86-digit number).
a(37) = a(38) = 0;
a(39) <= 1134572901070399771884918212890624;
a(41) <= 350847983^40 (a 342-digit number). (End)

			For n = 5; a(5) = 81 because 81 is the smallest number m such that tau(m) = tau(81) = 5 and tau(82) = tau(m) - 1 = 4.

Cf. A000005, A068208, A343144.

Magma
```
Ax:=func; [Ax(n): n in [1..5]]
```

a(6)-a(8) from Jon E. Schoenfield, Apr 20 2022
a(9)-a(10), a(16), a(21)-a(22), a(28), a(33) from Jaroslav Krizek, Apr 20 2022
Remaining terms through a(34) from Jon E. Schoenfield, Apr 30 2022
a(35) from Jinyuan Wang, May 21 2022

A368735 Table read by ascending antidiagonals: A(n,m) is the smallest number k such that k and k+1 have the n-th and m-th prime signatures, respectively, or -1 if no such k exists.

Original entry on oeis.org

-1, -1, 1, -1, 2, -1, -1, 4, 3, -1, -1, 6, -1, 5, -1, -1, -1, -1, 9, 7, -1, -1, 12, 8, 14, -1, 11, -1, -1, 16, -1, -1, 26, 49, -1, -1, -1, 40, -1, 20, -1, 51, -1, 23, -1, -1, 30, 24, 81, 124, 27, 15, -1, 29, -1, -1, -1, -1, 54, -1, 44, -1, 39, 169, 31, -1

Offset: 1

Jon E. Schoenfield, Jan 04 2024

			A(6,10) = 242 because 242 is the smallest number k of the form p^2 * q (the 6th prime signature; see A025487) such that k+1 is of the form r^5 (the 10th prime signature): 242 = 2 * 11^2 and 243 = 3^5.
A(2,7) = -1 because there exists no number k such that k is a prime (the 2nd prime signature) and k+1 is the fourth power of a prime (the 7th prime signature). (If k+1 = q^4 for some prime q, then k = (q-1)*(q+1)*(q^2+1), which cannot be a prime.)
The table below gives additional terms.
.
  n\m|  1   2  3   4      5    6  7        8      9      10   11       12
  ---+-------------------------------------------------------------------
   1 | -1   1 -1  -1     -1   -1 -1       -1     -1      -1   -1       -1
   2 | -1   2  3   5      7   11 -1       23     29      31   -1       47
   3 | -1   4 -1   9     -1   49 -1       -1    169      -1   -1    57121
   4 | -1   6 -1  14     26   51 15       39     65      -1   35      111
   5 | -1  -1  8  -1     -1   27 -1      343   2197      -1   -1       -1
   6 | -1  12 -1  20    124   44 -1      188    153     242   99      175
   7 | -1  16 -1  81     -1   -1 -1       -1 130321      -1   -1       -1
   8 | -1  40 24  54     -1  152 -1      135    104      -1   -1     1647
   9 | -1  30 -1 105 205378  170 -1      231    230   16806  195      890
  10 | -1  -1 -1  32     -1  243 -1       -1   3125      -1   -1       -1
  11 | -1  36 -1 225     -1 1444 -1 69189124    441      -1   -1 96393124
  12 | -1 112 48 176   4912  368 80      567    272 1419856 6723     2511

Cf. A025487, A046523, A343144.

A(n,m) = min_{ k : A046523(k) = A025487(n) AND A046523(k+1) = A025487(m) }, or -1 if no such k exists.

cp's OEIS Frontend

A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.

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A353032 a(n) is the smallest number m with n divisors such that m+1 has n-1 divisors, or 0 if no such number exists.

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A368735 Table read by ascending antidiagonals: A(n,m) is the smallest number k such that k and k+1 have the n-th and m-th prime signatures, respectively, or -1 if no such k exists.

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