A124477 -id:A124477 - cp's OEIS frontend

A145040 Primes p such that 2^p-1 is prime and congruent to 31 mod 5!.

5, 13, 17, 61, 89, 521, 2281, 3217, 4253, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 132049, 859433, 1398269, 2976221, 3021377, 6972593, 13466917, 30402457, 32582657, 42643801, 43112609, 57885161

Offset: 1

Artur Jasinski, Sep 30 2008

Mersenne numbers (with the exception of the first one) are congruent to 7 or 31 mod 5!. This sequence is a subsequence of A000043.
Is this 2 together with the terms of A112634? - R. J. Mathar, Mar 18 2009
Yes. An odd index p > 2 will be congruent to either 1 or 3 mod 4. If it is 1, then 2^p = 2^(4k+1) will be congruent to 2 mod 5, to 0 mod 4, and to 2 mod 3. This completely determines 2^p (and hence 2^p - 1) mod 5!. The other case, when p is congruent to 3 mod 4, will make 2^p congruent to 3 mod 5, to 0 mod 4, and to 2 mod 3. This leads to the other (distinct) value of 2^p mod 5!. This proves that this sequence is just A112634 without the initial term 2. - Jeppe Stig Nielsen, Jan 02 2018
From Jinyuan Wang, Nov 24 2019: (Start)
2^a(n) - 1 is congruent to 1 mod 5 since a(n) is congruent to 1 mod 4, so 5^(2^(a(n)-1) - 1) == (5, 2^a(n) - 1) == (2^a(n) - 1, 5)*(-1)^(2^a(n) - 1) == 1 (mod 2^a(n) - 1), where (m,p) is the Legendre symbol.
Conjecture: For n > 1, the Mersenne number M(n) = 2^n - 1 is in this sequence iff 5^M(n-1) == 1 (mod M(n)). (End)

Chris K. Caldwell, The largest known primes. - _R. J. Mathar_, Jul 31 2009

Cf. A000043, A000668, A112634, A124477, A139484, A145038, A112633, A145041, A145042.

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 5! ] == 31, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a
Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 120] == 32 &] (* Amiram Eldar, Oct 19 2024 *)

isok(p) = isprime(p) && isprime(q=2^p-1) && ((q % 120)==31); \\ Michel Marcus, Jan 06 2018

a(n) = A112634(n+1). - Jeppe Stig Nielsen, Jan 02 2018

42643801 inserted by R. J. Mathar, Jul 31 2009

a(28) from Amiram Eldar, Oct 19 2024

A135984 a(n) = 24(prime(n))+7.

Original entry on oeis.org

55, 79, 127, 175, 271, 319, 415, 463, 559, 703, 751, 895, 991, 1039, 1135, 1279, 1423, 1471, 1615, 1711, 1759, 1903, 1999, 2143, 2335, 2431, 2479, 2575, 2623, 2719, 3055, 3151, 3295, 3343, 3583, 3631, 3775, 3919, 4015, 4159, 4303, 4351, 4591, 4639, 4735

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Cf. A107006, A124477, A135657, A135982, A135983.

Mathematica
```
Table[24*Prime[n] + 7, {n, 1, 100}]
```

A135985 Prime numbers of the form 24*p + 7 where p is prime.

Original entry on oeis.org

79, 127, 271, 463, 751, 991, 1039, 1279, 1423, 1471, 1759, 1999, 2143, 2719, 3343, 3583, 3631, 3919, 4159, 4591, 4639, 4783, 5503, 5743, 5791, 7039, 7951, 8623, 9103, 9199, 9343, 9631, 10111, 10399, 10639, 11071, 11119, 11503, 12511

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A107006, A124477, A135657, A135982, A135983, A135984.

select(t -> isprime(t) and isprime((t-7)/24), [seq(p,p=7..20000,24)]); # Robert Israel, Oct 16 2018

a = {}; Do[If[PrimeQ[24(Prime[n]) + 7], AppendTo[a, 24(Prime[n]) + 7]], {n, 1, 100}]; a

A233008 p mod 24, where p is such that 2^p - 1 is prime (see Mersenne primes, A000043).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 7, 13, 17, 11, 7, 17, 7, 7, 19, 1, 1, 5, 7, 17, 5, 5, 17, 5, 1, 1, 11, 7, 1, 19, 23, 17, 19, 5, 5, 17, 17, 13, 19, 7, 23, 1, 17, 11, 1, 17, 17

Offset: 1

Freimut Marschner, Dec 03 2013

Chris K. Caldwell, Mersenne Primes.
Chris K. Caldwell, Recent Mersenne primes.
Walter Wegscheider, Liste der Mersenne-Primzahlen, Pädagogische Hochschule für Niederösterreich (in German).

Cf. A000043, A000668, A124477, A126043-A126059, A233009.

Mod[#, 24] &@ MersennePrimeExponent@ Range@ 45 (* Michael De Vlieger, Jul 22 2018 *)

a(n) = A000043(n) mod 24.

a(46)-a(47) corrected and a(48) removed by Gord Palameta, Jul 21 2018

a(48) from Amiram Eldar, Oct 15 2024

A135658 Nonprimes of the form 4x^2-4xy+7y^2.

Original entry on oeis.org

4, 15, 16, 24, 28, 36, 40, 55, 60, 63, 64, 87, 88, 96, 100, 112, 124, 132, 135, 144, 159, 160, 168, 175, 196, 216, 220, 231, 232, 240, 247

Offset: 1

Artur Jasinski, Nov 25 2007

nonn

Because 4x^2-4*x*y+7*y^2 = (2*x-y)^2+6*y^2, this is a subsequence of A002481. - R. J. Mathar, Jan 18 2021

Cf. A107006, A124477, A135657.

Do[Do[w = 4x^2 - 4x y + 7y^2; If[w > 0, If[PrimeQ[w],[null], AppendTo[a, w]]], {x, 0, 100}], {y, 0, 100}]; Union[a]

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A145040 Primes p such that 2^p-1 is prime and congruent to 31 mod 5!.

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A135984 a(n) = 24(prime(n))+7.

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A135985 Prime numbers of the form 24*p + 7 where p is prime.

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A233008 p mod 24, where p is such that 2^p - 1 is prime (see Mersenne primes, A000043).

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A135658 Nonprimes of the form 4x^2-4xy+7y^2.

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