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

1,2

Comments

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.

Links

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

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A301915, A301916, A301919.

Sequence in context: A281148 A284791 A281111 * A327312 A047243 A277094

Adjacent sequences: A301914 A301915 A301916 * A301918 A301919 A301920

Keyword

nonn

Author

Luke W. Richards, Mar 28 2018

Status

approved