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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.

%I A283620 #21 Oct 22 2023 11:32:27
%S A283620 2,-1,20,42,5,39,272,342,253,812,930,666,328,1806,1081,2756,1711,610,
%T A283620 1474,2485,876,6162,3403,7832,4656,10100,3502,5671,2943,12656,16002,
%U A283620 8515,18632,19182,22052,7550,12246,26406,13861,29756,15931,8145,18145,3088,38612,39402,44310
%N 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.
%C A283620 a(2) is -1, because 3^n-1 cannot be divisible by prime(2)=3.
%C A283620 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).
%C A283620 If n <> 2, then a(n) = A062117(n) if 3^A062117(n) == 1 (mod prime(n)^2), or
%C A283620 prime(n)*A062117(n) if not. - _Robert Israel_, Mar 16 2017
%H A283620 Robert Israel, <a href="/A283620/b283620.txt">Table of n, a(n) for n = 1..10000</a> (first 100 terms from Anton Mosunov)
%p A283620 subs(FAIL=-1,[seq(numtheory:-order(3, ithprime(i)^2), i=1..100)]); # _Robert Israel_, Mar 16 2017
%t A283620 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 *)
%o A283620 (PARI) a(n) = if (n == 2, -1, k = 1; p = prime(n); while((3^k-1) % p^2, k++); k;);
%Y A283620 Cf. A024023, A062117, A249025, A283454.
%K A283620 sign,look
%O A283620 1,1
%A A283620 _Michel Marcus_, Mar 12 2017