A112633 -id:A112633 - cp's OEIS frontend

A145039 Duplicate of A112633.

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

Offset: 1

dead

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

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

A145042 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 127 mod 6!.

Original entry on oeis.org

7, 19, 31, 127, 607, 1279, 2203, 4423, 110503, 216091, 1257787, 20996011, 24036583

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 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! ] == 127, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a

A112634 Mersenne prime indices that are not Gaussian primes.

Original entry on oeis.org

2, 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

Jorge Coveiro, Dec 27 2005

57885161, 74207281 and 77232917 are in this sequence as well. - Ivan Panchenko, Apr 13 2018
82589933 is in the sequence as well. - David Benjamin, Mar 30 2022
136279841 is in the sequence. - David Benjamin, Nov 11 2024
Other than the term 2, primes p (A000043) such that 2^p - 1 is prime (A000668) and congruent to 31 mod 120. - Jianing Song, Nov 18 2024

Chris K. Caldwell, The largest known primes.

Cf. A000043, A112633, A002313.

Select[MersennePrimeExponent[Range[48]], Mod[#, 4] != 3 &] (* Amiram Eldar, Oct 18 2024 *)

is(n)=n%4 < 3 && isprime(n) && isprime(2^n-1) \\ Charles R Greathouse IV, Nov 28 2016

A000043 INTERSECT A002313. - R. J. Mathar, Oct 06 2008

A000043 SET-MINUS A112633.

Edited by R. J. Mathar, Oct 06 2008
a(26)-a(28) from Ivan Panchenko, Apr 13 2018
a(29) from Amiram Eldar, Oct 18 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

A027206 Numbers m such that (1+i)^m + i is a Gaussian prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 11, 14, 16, 19, 38, 47, 62, 79, 151, 163, 167, 214, 239, 254, 283, 367, 379, 1214, 1367, 2558, 4406, 8846, 14699, 49207, 77291, 160423, 172486, 221006, 432182, 1513678, 2515574

Offset: 1

Ed Pegg Jr, Aug 07 2002

nonn

Equivalently, either (1+i)^m + i times its conjugate is an ordinary prime, or m == 2 (mod 4) and 2^(m/2) + (-1)^((m-2)/4) is an ordinary prime.

Let z = (1+i)^m + i. If z is not pure real or pure imaginary, then z is a Gaussian prime if the product of z and its conjugate is a rational prime. That product is 1 + 2^m + sin(m*Pi/4)*2^(1+m/2). z is imaginary when m=4k+2, in which case z has magnitude 2^(2k+1) + (-1)^k. These pure imaginary numbers are Gaussian primes when 2^(2k+1)-1 is a Mersenne prime and 2k+1 == 1 (mod 4); that is, when m is twice an odd number in A112633. - T. D. Noe, Mar 07 2011

Index entries for Gaussian integers and primes

Cf. A057429, A103329.

Select[Range[0,30000], PrimeQ[(1+I)^#+I, GaussianIntegers->True]&]

More terms from Mike Oakes, Aug 07 2002
Edited by Dean Hickerson, Aug 14 2002
0 prepended by T. D. Noe, Mar 07 2011

A112648 Integers k for which the k-th Mersenne prime exponent is a Gaussian prime (3 mod 4).

Original entry on oeis.org

2, 4, 7, 8, 11, 12, 14, 15, 16, 20, 28, 29, 31, 32, 34, 40, 41, 42, 45

Offset: 1

Jorge Coveiro, Dec 27 2005

nonn

			A000043(45) = 37156667 is congruent to 3 mod 4, so 45 is in this sequence.
A000043(46) = 42643801 is congruent to 1 mod 4, so 46 is not in this sequence.

Cf. A000043, A112633, A112634, A112649.

Select[Range@ 45, Mod[MersennePrimeExponent@ #, 4] == 3 &] (* Michael De Vlieger, Jul 22 2018 *)

Edited by Don Reble, Jan 25 2006

More terms from Gord Palameta, Jul 21 2018

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

A145039 Duplicate of A112633.

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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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A145042 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 127 mod 6!.

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A112634 Mersenne prime indices that are not Gaussian primes.

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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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A027206 Numbers m such that (1+i)^m + i is a Gaussian prime.

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A112648 Integers k for which the k-th Mersenne prime exponent is a Gaussian prime (3 mod 4).

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

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