A260072 -id:A260072 - cp's OEIS frontend

A260406 Numbers n such that (n-1)^2-1 divides 2^(n-1)-1.

1, 3, 5, 17, 37, 257, 457, 1297, 2557, 4357, 6481, 8009, 11953, 26321, 44101, 47521, 47881, 49681, 57241, 65537, 74449, 84421, 97813, 141157, 157081, 165601, 225457, 278497, 310591, 333433, 365941, 403901, 419711, 476737, 557041, 560737, 576721, 647089, 1011961

Offset: 1

M. F. Hasler, Jul 24 2015

nonn

The initial 1 is conventional.
647089 is the smallest composite number of this sequence (which makes it different from A081762).
The next composite number in this sequence is a(1000) = F_5 = 4294967297. - Robert G. Wilson v, Jul 25 2015
The Fermat numbers 2^2^k+1 = A000215(k) with k>1 are a subsequence of this sequence. I conjecture that they are equal to the intersection of this and A260407 (apart from the conventional 1), i.e., the numbers such that (n-1)^4-1 divides 2^(n-1)-1.

Robert G. Wilson v, Table of n, a(n) for n = 1..1598

Cf. A081762, A260072, A260407.

[n: n in [3..6*10^5] | (2^(n-1)-1) mod ((n-1)^2-1) eq 0]; // Vincenzo Librandi, Jul 26 2015

fQ[n_] := PowerMod[2, n - 1, (n - 1)^2 - 1] == 1; Select[ Range[3, 1200000], fQ] (* Robert G. Wilson v, Jul 25 2015 *)

forstep(n=1,1e7,2,Mod(2,(n-1)^2-1)^(n-1)==1&&print1(n","))

A260407 Numbers n such that (n-1)^2+1 divides 2^(n-1)-1.

Original entry on oeis.org

1, 17, 257, 8209, 65537, 649801, 1382401, 4294967297

Offset: 1

M. F. Hasler, Jul 24 2015

a(7) = 1382401 is the first composite number of this sequence (which makes it different from A260072).
The Fermat numbers 2^(2^k)+1 = A000215(k) with k>1 are a subsequence of this sequence. I conjecture that they are equal to the intersection of this and A260406 (except for the conventional 1).
Conjecture: also numbers n such that ((2^k)^(n-1)-1) == 0 mod ((n-1)^2+1) for all k >= 1. - Jaroslav Krizek, Jun 02 2016

Cf. A000215, A081762, A247165, A260072, A260406.

[n: n in [1..10^6] | (2^(n-1)-1) mod ((n-1)^2+1) eq 0 ]; // Vincenzo Librandi, Jul 25 2015

Join [{1},Select[Range[43*10^8],PowerMod[2,#-1,(#-1)^2+1]==1&]] (* Harvey P. Dale, Sep 07 2018 *)

forstep(n=1,1e7,2,Mod(2,(n-1)^2+1)^(n-1)==1&&print1(n","))

a(n) = A247165(n)+1.

cp's OEIS Frontend

A260406 Numbers n such that (n-1)^2-1 divides 2^(n-1)-1.

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