A188133 - OEIS

%I #32 Dec 09 2024 10:58:51

%S 43,487,547,571,883,1459,1663,1723,2539,3319,3511,4903,5107,5431,6199,

%T 6367,8011,8599,9007,9391,9511,10111,11119,11959,12379,12703,13291,

%U 13339,13999,14083,14551,14767,15187,15319,15859,15991,16183,16603,16747,17659,18427,19699

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

%C 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.

%C The equivalent sequence of prime indices is 14, 93, 101, 105, 153, 232, 261, 269, ....

%C 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

%H Amiram Eldar, <a href="/A188133/b188133.txt">Table of n, a(n) for n = 1..10000</a>

%F {p = A000040(i): 10*p+1 | A001348(i)}. - _R. J. Mathar_, Mar 21 2011

%t Select[Range[2*10^4], PrimeQ[#] && PowerMod[2, #, 10# + 1] == 1 &] (* _Amiram Eldar_, Nov 13 2019 *)

%t Select[Prime[Range[2500]],PowerMod[2,#,10#+1]==1&] (* _Harvey P. Dale_, Dec 08 2024 *)

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

%o (Python) from sympy import sieve

%o print([p for p in sieve[1:10000] if pow(2,p,10*p+1) == 1])

%o # _Karl-Heinz Hofmann_, Jan 27 2022

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

%Y Cf. A001348, A000040.

%K nonn

%O 1,1

%A _M. F. Hasler_, Mar 21 2011