A301919 -id:A301919 - cp's OEIS frontend

A301917 a(n) is the least k for which A301916(n) divides 3^k + 1.

1, 2, 3, 8, 9, 14, 15, 9, 4, 21, 26, 5, 11, 6, 39, 44, 24, 50, 17, 56, 63, 68, 69, 74, 25, 39, 81, 86, 8, 98, 99, 105, 111, 116, 60, 128, 134, 15, 140, 141, 146, 17, 158, 165, 84, 87, 176, 61, 93, 189, 194, 99, 200, 102, 73, 224, 114, 230, 231, 243, 83, 254

Offset: 1

Luke W. Richards, Mar 28 2018

nonn

This can be used to factor P-1 values for potential primes, P of the form 3^k+2.

A301915 can be used in conjunction with this sequence such that A301916 always divides 3^(a(n) + k*A301915(n)) + 1 for all nonnegative values of k.

			A301916(1) = 2 and the first value of k for which 3^k+1 is a multiple of 2 is k = 1, so a(1) = 1.
A301916(5) = 19 and the first value of k for which 3^k+1 is a multiple of 19 is k = 9, so a(5) = 9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A301915, A301916, A301919.

f:= proc(p) local t; t:= numtheory:-order(3,p); if t::even then t/2 else NULL fi end proc:
f(2):= 1:
map(f, [seq(ithprime(i),i=1..300)]); # Robert Israel, May 23 2018

f[p_] := Module[{t = MultiplicativeOrder[3, p]}, If[EvenQ[t],  t/2, Nothing]];
f[2] = 1;
f /@ Table[Prime[i], {i, 1, 300}] (* Jean-François Alcover, Feb 02 2023, after Robert Israel *)

lista(nn) = {for (n=1, nn, p = prime(n); if (p != 3, m = Mod(3, p); nb = znorder(m); for (k=1, nb, if (m^k == Mod(-1, p), print1(k, ", ")););););} \\ Michel Marcus, May 18 2018

a(n) = A301919(n+1) - 1 for n > 1.

A301918 Primes which divide numbers of the form 3^k+3.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 113, 127, 137, 139, 149, 151, 157, 163, 173, 193, 197, 199, 211, 223, 233, 241, 257, 269, 271, 281, 283, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 439

Offset: 1

Luke W. Richards, Mar 28 2018

nonn

Union of {3} and A301916, because 3^k + 3 = 3*(3^(k-1) + 1). [Comment edited by Jeppe Stig Nielsen, Jul 04 2020.]
Can be used to factor P+1 values where P is a potential prime of the form 3^k+2.
Is this 2 and 3 with A045318? - David A. Corneth, May 04 2018
No, it is not. Primes like 769, 1297, ... are also here but not in A045318. See A320481 for the explanation. - Jeppe Stig Nielsen, Jun 27 2020

			All values of 3^k+3 are multiples of 2, so 2 is in the sequence.
3^4+3 = 84, which is a multiple of 7, so 7 is in the sequence.

Cf. A045318, A301916, A301917, A301919.

cp's OEIS Frontend

A301917 a(n) is the least k for which A301916(n) divides 3^k + 1.

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