A145042 -id:A145042 - cp's OEIS frontend

A112633 Mersenne prime indices that are also Gaussian primes.

3, 7, 19, 31, 107, 127, 607, 1279, 2203, 4423, 86243, 110503, 216091, 756839, 1257787, 20996011, 24036583, 25964951, 37156667

Offset: 1

Jorge Coveiro, Dec 27 2005

Also, primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 7 mod 5!. - Artur Jasinski, Sep 30 2008. Proof that this is the same sequence, from Jeppe Stig Nielsen, Jan 02 2018: An odd index p>2 will be either 1 or 3 mod 4. If it is 1, then 2^p = 2^(4k+1) will be 2 mod 5, and be 0 mod 4, and be 2 mod 3. This completely determines 2^p (and hence 2^p - 1) mod 5!. The other case, when p is 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!.

Cf. A112634, A112648, A112649.

Cf. also A000043, A000668, A002145, A124477, A139484, A145039, A145040, 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! ] == 7, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a (* Artur Jasinski, Sep 30 2008 *)
Select[{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}, Mod[2^#-1,120]==7&] (* Harvey P. Dale, Nov 26 2013 *)
Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 120] == 8 &] (* Amiram Eldar, Oct 19 2024 *)

from itertools import count, islice
from sympy import isprime, prime
def A112633_gen(): # generator of terms
    return filter(lambda p: p&2 and isprime((1<A112633_list = list(islice(A112633_gen(),10)) # Chai Wah Wu, Mar 21 2023

The intersection of A000043 and A002145. - R. J. Mathar, Oct 06 2008

Edited by N. J. A. Sloane, Jan 06 2018

a(19) from Ivan Panchenko, Apr 12 2018

A145041 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 31 mod 6!.

Original entry on oeis.org

5, 17, 89, 521, 4253, 9689, 9941, 11213, 19937, 21701, 859433, 1398269, 2976221, 3021377, 6972593, 32582657, 43112609, 57885161

Offset: 1

Artur Jasinski, Sep 30 2008

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subsequence of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

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, 6! ] == 31, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a
Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 720] == 32 &] (* Amiram Eldar, Oct 19 2024 *)

a(18) from Amiram Eldar, Oct 19 2024

A145038 Numbers to which Mersenne primes 2^p-1 can be congruent mod k! (for k > 1).

Original entry on oeis.org

1, 3, 7, 31, 127, 271, 607, 2047, 3151, 8191, 10111, 40447, 42367, 48511, 50431, 80767, 88831, 90751, 121087, 131071, 161407, 163327, 169471, 171391, 201727, 209791, 211711, 243967, 250111, 282367, 290431, 292351, 322687, 324607, 332671

Offset: 1

Artur Jasinski, Sep 30 2008

nonn

All Mersenne primes are congruent to 1 mod 2!, 1 mod 3! (with the exception of the first one), 7 mod 4! (with the exception of the first one), 7 mod 5! (see A112633), or 31 mod 5! (see A145040).

Cf. A000668, A124477, A139484, A112633, A145040, A145041, A145042.

A145044 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 271 mod 6!.

Original entry on oeis.org

13, 61, 2281, 3217, 23209, 44497, 132049, 13466917, 30402457, 42643801

Offset: 1

Artur Jasinski, Sep 30 2008

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subsequence of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

Select[MersennePrimeExponent[Range[47]], PowerMod[2, #, 6!] == 272 &] (* Amiram Eldar, Mar 22 2020 *)

a(10) from Amiram Eldar, Mar 22 2020

A145045 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 607 mod 6!

Original entry on oeis.org

107, 86243, 756839, 25964951, 37156667

Offset: 1

Artur Jasinski, Sep 30 2008

nonn

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subset of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

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, 6! ] == 607, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a (*Artur Jasinski*)

Comment rewritten by Harvey P. Dale, Sep 02 2023

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

Original entry on oeis.org

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

cp's OEIS Frontend

A112633 Mersenne prime indices that are also Gaussian primes.

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A145041 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 31 mod 6!.

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A145038 Numbers to which Mersenne primes 2^p-1 can be congruent mod k! (for k > 1).

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A145044 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 271 mod 6!.

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A145045 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 607 mod 6!

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

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