A188130 -id:A188130 - cp's OEIS frontend

A122095 Primes p for which 8*p+1 divides 2^p-1.

11, 29, 179, 239, 431, 761, 857, 941, 1367, 1667, 1871, 1877, 2411, 2837, 3041, 3119, 3329, 3347, 3767, 4289, 5021, 5087, 5231, 5261, 5717, 5861, 6449, 6917, 6959, 7079, 7211, 7919, 8429, 8741, 8867, 9341, 9461, 9851, 10211, 10979, 12107, 12437, 12479

Offset: 1

J. Lowell, Oct 17 2006

nonn

The first 962 terms, all those with n<500000, are also in A023228. - R. J. Mathar, Oct 20 2006

All terms are in A023228, i.e., such that 8p+1 is prime, since a divisor of 8p+1 would also divide M(p)=A000225(p) and thus be of the form 2kp+1, but it is easily checked that 8p+1 cannot be a multiple of 2p+1 (nor of 4p+1 or 6p+1, of course). - M. F. Hasler, Mar 21 2011

			29 is in this sequence because 2^29-1 is divisible by 8 * 29 + 1 = 233.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000225, A002515, A188130.

isA122095 := proc(n) RETURN( isprime(n) and ( (2^n-1) mod (8*n+1)) = 0 ) ; end: n := 1 : for a from 2 to 500000 do if isA122095(a) then print(n,a) ; n := n+1 ; fi ; od ; # R. J. Mathar, Oct 20 2006

Select[Prime[Range[1500]],Divisible[2^#-1,8#+1]&] (* Harvey P. Dale, Dec 18 2012 *)
Select[Prime[Range[1500]],PowerMod[2,#,8#+1]==1&] (* Harvey P. Dale, May 28 2015 *)

forprime( p=1,1e4, Mod(2,p*8+1)^p-1 || print1(p, ", "))

More terms from R. J. Mathar, Oct 20 2006

A188133 Primes p such that 10p+1 divides 2^p-1.

Original entry on oeis.org

43, 487, 547, 571, 883, 1459, 1663, 1723, 2539, 3319, 3511, 4903, 5107, 5431, 6199, 6367, 8011, 8599, 9007, 9391, 9511, 10111, 11119, 11959, 12379, 12703, 13291, 13339, 13999, 14083, 14551, 14767, 15187, 15319, 15859, 15991, 16183, 16603, 16747, 17659, 18427, 19699

Offset: 1

M. F. Hasler, Mar 21 2011

nonn

It is known that divisors of M(p)=2^p-1 are of the form 2kp+1. For k=1, these are the Lucasian primes A002515, for k=2 there are no such divisors, for k=3 these divisors are listed in A188130 and for k=4 in A122095.
The equivalent sequence of prime indices is 14, 93, 101, 105, 153, 232, 261, 269, ....
If k == 2 (mod 4), there are no such divisors in general and here there are no smaller k's than k = 5. - Karl-Heinz Hofmann, Jan 27 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A350702 (k = 7).

Cf. A001348, A000040.

Select[Range[2*10^4], PrimeQ[#] && PowerMod[2, #, 10# + 1] == 1 &] (* Amiram Eldar, Nov 13 2019 *)
Select[Prime[Range[2500]],PowerMod[2,#,10#+1]==1&] (* Harvey P. Dale, Dec 08 2024 *)

forprime(p=1,1e5, Mod(2,p*10+1)^p-1 || print1(p", "))

from sympy import sieve
print([p for p in sieve[1:10000] if pow(2,p,10*p+1) == 1])
# Karl-Heinz Hofmann, Jan 27 2022

{p = A000040(i): 10*p+1 | A001348(i)}. - R. J. Mathar, Mar 21 2011

A350702 Primes p such that 14*p + 1 divides 2^p - 1.

Original entry on oeis.org

929, 1433, 2393, 2609, 2657, 4373, 4793, 6029, 7529, 10133, 10433, 10949, 10973, 13049, 13109, 16829, 18869, 20873, 22349, 23417, 24137, 26717, 27737, 27893, 28433, 28517, 30977, 33809, 33857, 37217, 38189, 38237, 39209, 39749, 41453, 41813, 42569, 43313, 43613

Offset: 1

Karl-Heinz Hofmann, Jan 27 2022

nonn

Known divisors of Mersenne(p) (2^p-1 or Mp for short) are of the form 2*k*p+1. See crossrefs for other k's. If k == 2 (mod 4), there are no such divisors in general. Here k is 14/2 = 7.

			See LINKS for example of a(13).

Karl-Heinz Hofmann, Mersenne(p) table with k = 7 and, if they exist, additional k < 7 plus their corresponding factors.
mersenne.ca, a(13) = M10973 Mersenne number exponent details.
mersenne.ca, a(1..995) at GIMPS with k = 7 and Exponent 3..2000000.

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A188133 (k = 5).

Cf. A000040, A001348.

Select[Range[45000], PrimeQ[#] && PowerMod[2, #, 14*# + 1] == 1 &] (* Amiram Eldar, Jan 27 2022 *)

forprime(p=1, 1e6, if (Mod(2, p*14+1)^p==1, print1(p,", ")))

from sympy import sieve
print([p for p in sieve[1:1000000] if pow(2, p, 14*p+1) == 1])

{p = A000040(i): 14*p+1 | A001348(i)}.

A038844 Numbers k for which 6*k+1 divides 2^k-1.

Original entry on oeis.org

5, 21, 37, 72, 73, 76, 100, 121, 153, 221, 233, 237, 245, 276, 288, 292, 296, 300, 305, 333, 336, 341, 348, 352, 357, 380, 381, 397, 445, 448, 461, 465, 472, 492, 545, 557, 565, 576, 577, 601, 605, 637, 648, 657, 676, 688, 692, 696, 737, 752, 753, 761, 776

Offset: 1

Bill Gosper

nonn

Apart from 5, all terms are in A045762, numbers k such that 2^k-1 is not prime. - Michel Marcus, Nov 12 2014

6*k + 1 is not necessarily a prime for k being a term of this sequence. - Jianing Song, Jun 20 2025

			For n=5, 2^5-1=31 is divisible by 6*5+1=31.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A045762, A188130 (subsequence of primes).

Select[Range[800],PowerMod[2,#,6#+1]==1&] (* Harvey P. Dale, Oct 24 2017 *)

select( {is_A038844(n)=Mod(2,n*6+1)^n==1}, [1..999]) \\ M. F. Hasler, Aug 17 2021

More terms from Michel Marcus, Nov 12 2014

A188131 Primes p == 1 (mod 4) such that 6p+1 is prime.

Original entry on oeis.org

5, 13, 17, 37, 61, 73, 101, 137, 173, 181, 233, 241, 257, 277, 293, 313, 373, 397, 461, 557, 577, 593, 601, 641, 653, 661, 761, 773, 797, 853, 937, 941, 1013, 1033, 1061, 1117, 1193, 1201, 1321, 1361, 1381, 1433, 1453, 1481, 1553, 1613, 1693, 1733, 1777, 1873, 1973, 1993

Offset: 1

M. F. Hasler, Mar 21 2011

nonn

Contains A188130 as a subsequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A188130.

Select[Range[1, 2000, 4], PrimeQ[#] && PrimeQ[6# + 1] &] (* Amiram Eldar, Nov 13 2019 *)

forprime(p=1,1e5, p%4==1 & isprime(p*6+1) & print1(p", "))

A350703 a(n) is the least integer k such that (2nk+1) | (2^k-1).

Original entry on oeis.org

3, 18, 5, 9, 15, 50, 40, 16, 7, 156, 60, 25, 180, 102, 113, 81, 10, 50, 29, 159, 51, 56, 24, 36, 47, 90, 337, 72, 55, 106, 33, 102, 780, 28, 117, 25, 155, 540, 60, 104, 223, 1012, 168, 180, 91, 540, 3132, 47, 510, 412, 154, 45, 80, 432, 201, 36, 90, 144, 97, 53, 279, 880

Offset: 1

Karl-Heinz Hofmann, Feb 03 2022

nonn

The formula 2nk+1 is used to find trivial factors of Mersenne(p). Here it is used for all exponents (prime exponents and not prime exponents).
Mersenne primes of A000043 can be found in this sequence too (except for 2). E.g.: a(1, 3, 9, 315, 3855, 13797) = A000043(2..7).
If n mod 4 = 2 then a(n) must be composite.

			a(5) = 15: 2^15 - 1 = 32767; 2*5*15 + 1 = 151; 32767 mod 151 = 0, and there are no numbers < 15 which satisfy the requirement for n = 5.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000043, A002515, A188130, A122095, A188133, A350702.

a[n_] := Module[{k = 1}, While[PowerMod[2, k, 2*n*k + 1] != 1, k++]; k];  Array[a, 62] (* Amiram Eldar, Feb 03 2022 *)

a(n) = my(k=1); while (Mod(2, 2*n*k+1)^k != 1, k++); k; \\ Michel Marcus, Feb 03 2022

def A350703(k,expo):
    while pow(2, expo, 2*k*expo+1) != 1: expo += 1
    return expo
print([A350703(k,1) for k in range(1, 63)])

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A122095 Primes p for which 8*p+1 divides 2^p-1.

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A188133 Primes p such that 10p+1 divides 2^p-1.

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A350702 Primes p such that 14*p + 1 divides 2^p - 1.

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A038844 Numbers k for which 6*k+1 divides 2^k-1.

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A188131 Primes p == 1 (mod 4) such that 6p+1 is prime.

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A350703 a(n) is the least integer k such that (2*n*k+1) | (2^k-1).

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A350703 a(n) is the least integer k such that (2nk+1) | (2^k-1).