A350702 -id:A350702 - cp's OEIS frontend

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

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

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

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

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

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