A283620 a(n) is the least exponent k such that 3^k-1 is divisible by prime(n)^2, or -1 if no such k exists.

2, -1, 20, 42, 5, 39, 272, 342, 253, 812, 930, 666, 328, 1806, 1081, 2756, 1711, 610, 1474, 2485, 876, 6162, 3403, 7832, 4656, 10100, 3502, 5671, 2943, 12656, 16002, 8515, 18632, 19182, 22052, 7550, 12246, 26406, 13861, 29756, 15931, 8145, 18145, 3088, 38612, 39402, 44310

Offset: 1

Author

Michel Marcus, Mar 12 2017

Keywords

• sign
• look

Comments

a(2) is -1, because 3^n-1 cannot be divisible by prime(2)=3.

For some terms, prime(n)^2 is also the least square of prime which divides 3^a(n)-1. This is the case for n=1, 5, 6, ..., that is, p=2, 11, 13, ... (see A283454).

If n <> 2, then a(n) = A062117(n) if 3^A062117(n) == 1 (mod prime(n)^2), or

prime(n)*A062117(n) if not. - Robert Israel, Mar 16 2017

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 100 terms from Anton Mosunov)

Crossrefs

Cf. A024023, A062117, A249025, A283454.

Programs

Maple

subs(FAIL=-1,[seq(numtheory:-order(3, ithprime(i)^2), i=1..100)]); # Robert Israel, Mar 16 2017

Mathematica

Join[{2,-1},Table[Module[{k=1},While[PowerMod[3,k,Prime[n]^2]!=1,k++];k],{n,3,50}]] (* Harvey P. Dale, Oct 22 2023 *)

PARI

a(n) = if (n == 2, -1, k = 1; p = prime(n); while((3^k-1) % p^2, k++); k;);