A350703 - OEIS

%I #32 Feb 16 2022 13:20:49

%S 3,18,5,9,15,50,40,16,7,156,60,25,180,102,113,81,10,50,29,159,51,56,

%T 24,36,47,90,337,72,55,106,33,102,780,28,117,25,155,540,60,104,223,

%U 1012,168,180,91,540,3132,47,510,412,154,45,80,432,201,36,90,144,97,53,279,880

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

%C 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).

%C 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).

%C If n mod 4 = 2 then a(n) must be composite.

%H Karl-Heinz Hofmann, <a href="/A350703/b350703.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

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

%o (Python)

%o def A350703(k,expo):

%o     while pow(2, expo, 2*k*expo+1) != 1: expo += 1

%o     return expo

%o print([A350703(k,1) for k in range(1, 63)])

%o (PARI) a(n) = my(k=1); while (Mod(2, 2*n*k+1)^k != 1, k++); k; \\ _Michel Marcus_, Feb 03 2022

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

%K nonn

%O 1,1

%A _Karl-Heinz Hofmann_, Feb 03 2022