A283454 -id:A283454 - cp's OEIS frontend

A283453 The smallest square referenced in A249025 (Numbers k such that 3^k - 1 is not squarefree).

4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 169, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 4, 4, 4, 4, 121, 4, 169, 4, 4, 4, 4, 121

Offset: 1

Robert Price, Mar 07 2017

nonn

			A249025(3)=5, 3^5-1 = 242 = 2*11*11. 242 is not squarefree the square being 11*11 = 121.

Amiram Eldar, Table of n, a(n) for n = 1..407

Cf. A249025, A283454.

psq[n_] := If[(f = Select[FactorInteger[n], Last[#] > 1 &]) == {}, 1, f[[1, 1]]^2]; psq /@ Select[3^Range[100] - 1, !SquareFreeQ[#] &] (* Amiram Eldar, Feb 12 2021 *)

lista(nn) = {for (n=1, nn, if (!issquarefree(k = 3^n-1), f = factor(k/core(k)); vsq = select(x->((x%2) == 0), f[,2], 1); print1(f[vsq[1], 1]^2, ", ");););} \\ Michel Marcus, Mar 11 2017

a(n) = A283454(n)^2. - Amiram Eldar, Feb 12 2021

More terms from Michel Marcus, Mar 11 2017

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.

Original entry on oeis.org

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

Michel Marcus, Mar 12 2017

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

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

Cf. A024023, A062117, A249025, A283454.

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

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 *)

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

cp's OEIS Frontend

A283453 The smallest square referenced in A249025 (Numbers k such that 3^k - 1 is not squarefree).

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