A081762 -id:A081762 - cp's OEIS frontend

A247219 Positive numbers m such that m^2 - 1 divides 2^m - 1.

2, 4, 16, 36, 256, 456, 1296, 2556, 4356, 6480, 8008, 11952, 26320, 44100, 47520, 47880, 49680, 57240, 65536, 74448, 84420, 97812, 141156, 157080, 165600, 225456, 278496, 310590, 333432, 365940, 403900, 419710, 476736, 557040, 560736, 576720, 647088, 1011960, 1033056, 1204560, 1206180

Offset: 1

Juri-Stepan Gerasimov, Nov 26 2014

nonn

Contains all numbers of the form m = A001146(k) = 2^2^k, k >= 0; and those with k > 1 seem to form the intersection with A247165. - M. F. Hasler, Jul 25 2015

			2 is in this sequence because 2^2 - 1 = 3 divides 2^2 - 1 = 3.

Chai Wah Wu, Table of n, a(n) for n = 1..105

Cf. A081762.

[n: n in [2..122222] | Denominator((2^n - 1)/(n^2 - 1)) eq 1];

Select[Range[10^4], Divisible[2^# - 1, #^2 - 1] &] (* Alonso del Arte, Nov 26 2014 *)
Select[Range[2,121*10^4],PowerMod[2,#,#^2-1]==1&] (* Harvey P. Dale, Sep 08 2021 *)

isok(n) = ((2^n - 1) % (n^2 - 1)) == 0; \\ Michel Marcus, Nov 26 2014

forstep(n=0,1e8,2, Mod(2,n^2-1)^n-1 || print1(n", ")) \\ M. F. Hasler, Jul 25 2015

from gmpy2 import powmod
A247219_list = [n for n in range(2,10**7) if powmod(2,n,n*n-1) == 1]
# Chai Wah Wu, Dec 03 2014

Corrected a(24) by Chai Wah Wu, Dec 03 2014

A249751 Numbers m such that m - 2 divides m^m + 2.

Original entry on oeis.org

3, 4, 7, 8, 67, 260, 379, 1191, 1471, 5076, 25807, 58591, 103780, 134947, 137347, 170587, 203236, 272611, 285391, 420211, 453748, 538735, 540856, 592411, 618451, 680707, 778807, 1163067, 1306936, 1520443, 1700947, 1891336, 2099203, 2831011, 3481960, 4020031

Offset: 1

Juri-Stepan Gerasimov, Dec 05 2014

nonn

			3 is in this sequence because (3^3 + 2)/(3 - 2) = 29 is an integer.

Chai Wah Wu, Table of n, a(n) for n = 1..160

Cf. A081762, A242787, A213382, A252041.

[n: n in [3..10000] | Denominator((n^n+2)/(n-2)) eq 1];

fQ[n_] := Mod[ PowerMod[ n, n, n - 2] + 2, n - 2] == 0; Select[ Range@ 4100000, fQ] (* Robert G. Wilson v, Dec 19 2014 *)

A249751_list = [n for n in range(3,10**7) if n==3 or pow(n,n,n-2) == n-4]
# Chai Wah Wu, Dec 06 2014

More terms from Chai Wah Wu, Dec 06 2014

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

Original entry on oeis.org

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","))

A260072 Primes p such that (p-1)^2+1 divides 2^(p-1)-1.

Original entry on oeis.org

17, 257, 8209, 65537, 649801

Offset: 1

Jaroslav Krizek, Jul 22 2015

a(6), if it exists, is larger than 1.7*10^12. - Giovanni Resta, Jul 23 2015

N = 1382401 is the smallest composite number such that (n-1)^2+1 divides 2^(n-1)-1, cf. A260407; see also A081762 and A260406. The sequence contains all Fermat primes 2^2^k+1 > 5 (A019434). - M. F. Hasler, Jul 24 2015

			17 is in this sequence because (17 - 1)^2 + 1 = 257 divides 2^(17 - 1) - 1 = 65535; 65535 / 257 = 255.

Cf. A081762 (primes p such that (p-1)^2 - 1 divides 2^(p-1) - 1).

[n: n in [1..2000000] | IsPrime(n) and (2^(n-1)-1) mod ((n-1)^2 + 1) eq 0];

fQ[n_] := PowerMod[2, n-1, (n-1)^2 + 1] == 1; p = 2; lst = {}; While[p < 10^9, If[ fQ@ p, AppendTo[lst, p]]; p = NextPrime@ p] (* Robert G. Wilson v, Jul 24 2015 *)

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.

A337846 Odd integers k such that 2^((k-1)/2) == 1 (mod k*(k-2)).

Original entry on oeis.org

17, 257, 457, 1297, 6481, 11953, 26321, 47521, 47881, 49681, 65537, 74449, 157081, 165601, 278497, 333433, 476737, 557041, 560737, 576721, 1033057, 1266841, 1329337, 1463617, 1468897, 2291041, 2422201, 2754481, 2851633, 2969137, 3255281

Offset: 1

Benoit Cloitre, Sep 26 2020

nonn

Computed terms are prime. Is this a possible primality test or are there pseudo primes? Terms are of the form 8k+1.

The Fermat number F(5) = A000215(5) = 4294967297 = 641*6700417 is the smallest composite counterexample. - Hugo Pfoertner, Sep 26 2020

Cf. A081762, A337818.

Select[Range[3, 10^6, 2], PowerMod[2, (# - 1)/2, #*(# - 2)] == 1 &] (* Amiram Eldar, Sep 26 2020 *)

is(n) = n%2 && n>=3 && Mod(2, n*(n-2))^((n-1)/2) == 1

A337847 Odd integers k such that 3^((k-1)/2) == 1 (mod k*(k-2)).

Original entry on oeis.org

457, 1297, 6481, 14401, 26497, 44101, 47521, 47881, 165601, 225457, 446881, 560737, 576721, 677041, 1037857, 1049941, 1649341, 1903981, 1934137, 2291041, 3990601, 4110121, 4262161, 4663297, 4736341, 5293081, 5317057, 5372929, 6410497, 6535681, 6651361, 8122501

Offset: 1

Benoit Cloitre, Sep 26 2020

nonn

Computed terms are prime. Is this a possible primality test or are there pseudo primes? Terms are of the form 12k+1.

Cf. A081762, A337818.

Select[Range[3, 10^6, 2], PowerMod[3, (# - 1)/2, #*(# - 2)] == 1 &] (* Amiram Eldar, Sep 26 2020 *)

is(n) = n%2 && n>=3 && Mod(3, n*(n-2))^((n-1)/2) == 1

More terms from Amiram Eldar, Sep 26 2020

cp's OEIS Frontend

A247219 Positive numbers m such that m^2 - 1 divides 2^m - 1.

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A249751 Numbers m such that m - 2 divides m^m + 2.

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

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A260072 Primes p such that (p-1)^2+1 divides 2^(p-1)-1.

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

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A337846 Odd integers k such that 2^((k-1)/2) == 1 (mod k*(k-2)).

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A337847 Odd integers k such that 3^((k-1)/2) == 1 (mod k*(k-2)).

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Mathematica

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