A297362 -id:A297362 - cp's OEIS frontend

A345705 Numbers k such that (3^ord(3/2, k) - 2^ord(3/2, k))/k is a prime, where ord(3/2, k) is the multiplicative order of 3/2 (mod k).

13, 29, 35, 47, 53, 71, 95, 133, 263, 275, 485, 529, 773, 1009, 1261, 1559, 2711, 3767, 4009, 5275, 7613, 8645, 10295, 11605, 21311, 27755, 29927, 40565, 44519, 67135, 67849, 75335, 83333, 105469, 107185, 153557, 164365, 383705, 405623, 420341, 443105

Offset: 1

Amiram Eldar, Jun 24 2021

nonn

Numbers k such that gcd(k, 6) = 1 and if m is the least positive integer such that k divides 3^m - 2^m, then (3^m - 2^m)/k is a prime number.

The corresponding primes are 5, 71, 19, 2002867877, 29927, 29, 7, 5, ...

Wikipedia, Multiplicative order.
Wikipedia, Zsigmondy's theorem.

Cf. A297362, A297363, A320383.

ord[n_] := Module[{k = 1}, While[! Divisible[PowerMod[3, k, n] - PowerMod[2, k, n], n], k++]; k]; f[k_] := 3^k - 2^k; Select[Range[1000], CoprimeQ[6, #] && PrimeQ[f[ord[#]]/#] &]

13 is a term since ord(3/2, 13) = 4 and (3^4 - 2^4)/13 = 5 is a prime number.

cp's OEIS Frontend

A345705 Numbers k such that (3^ord(3/2, k) - 2^ord(3/2, k))/k is a prime, where ord(3/2, k) is the multiplicative order of 3/2 (mod k).

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