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

1,1

Comments

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, ...

Links

Table of n, a(n) for n=1..41.

Wikipedia, Multiplicative order.

Wikipedia, Zsigmondy's theorem.

Formula

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

Mathematica

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[#]]/#] &]

Crossrefs

Cf. A297362, A297363, A320383.

Sequence in context: A319167 A088909 A340919 * A322388 A096451 A090690

Adjacent sequences: A345702 A345703 A345704 * A345706 A345707 A345708

Keyword

nonn

Author

Amiram Eldar, Jun 24 2021

Status

approved