A112634 -id:A112634 - 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

A103329 Numbers n such that (1+i)^n - i is a Gaussian prime.

Original entry on oeis.org

0, 3, 4, 5, 8, 10, 16, 26, 29, 34, 73, 113, 122, 157, 178, 241, 353, 457, 997, 1042, 3041, 4562, 6434, 8506, 10141, 19378, 19882, 22426, 27529

Offset: 1

T. D. Noe, Jan 31 2005

nonn

Note that A027206 and A057429 treat Gaussian primes of a similar form. The remaining case, (1+i)^n + 1, is a Gaussian prime for n=1,2,3,4 only.

Let z = (1+i)^n - 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^n - sin(n*Pi/4)*2^(1+n/2). z is real when n=1. z is imaginary when n=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 = 3 (mod 4); that is, when n is twice an odd number in A112634. - T. D. Noe, Mar 07 2011

Cf. A027206 ((1+i)^n + i is a Gaussian prime), A057429 ((1+i)^n - 1 is a Gaussian prime).

fQ[n_] := PrimeQ[(1 + I)^n - I, GaussianIntegers -> True]; Select[ Range[0, 30000], fQ]

a(25)-a(29) from Robert G. Wilson v, Mar 02 2011.

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

A361563 Wagstaff numbers that are of the form 4*k + 1.

Original entry on oeis.org

5, 13, 17, 61, 101, 313, 701, 1709, 2617, 10501, 42737, 95369, 138937, 267017, 374321

Offset: 1

Jorge Coveiro, Mar 15 2023

15135397 is also in the sequence, but may not be the next term.

Jorge Coveiro, Possible 'Formula' for Wagstaff numbers, mersenneforum.org.

Cf. A000978 (Wagstaff numbers), A002144 (primes of form 4*k + 1), A112634, A361562.

from itertools import count, islice
from sympy import prime, isprime
def A361563_gen(): # generator of terms
    return filter(lambda p: not p&2 and isprime(((1<A361563_list = list(islice(A361563_gen(),7)) # Chai Wah Wu, Mar 21 2023

Intersection of A000978 and A002144.

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A112633 Mersenne prime indices that are also Gaussian primes.

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A103329 Numbers n such that (1+i)^n - 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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A361563 Wagstaff numbers that are of the form 4*k + 1.

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