A130058 Numbers m such that k = m*23^2 divides 3^(k-1) - 2^(k-1).

1, 67, 89, 133, 199, 331, 617, 793, 881, 5281, 8911, 11419, 13333, 22177, 23585, 26467, 35113, 35839, 38897, 40657, 44023, 54913, 65869, 67849, 70819, 92929, 105469, 107185, 114247, 124279, 144673, 153253, 159259, 185329, 196945, 225589

Offset: 1

Author

Alexander Adamchuk, May 04 2007

Keywords

• nonn

Comments

All prime p>3 divide 3^(p-1) - 2^(p-1). It appears that 1 and 529 = 23^2 are the only perfect squares in A073631. Most terms of A073631 are squarefree. First 50 nonsquarefree terms of A073631 are the multiples of 23^2.

Conjecture: All nonsquarefree terms of A073631 are the multiples of 23^2.

Prime terms are listed in A130059. Note that the many terms (namely, 1, 133, 793, 8911, 13333, 22177, 26467, 38897, 44023, 54913, 65869, ...) also belong to A073631.

Crossrefs

Cf. A073631, A001047, A130059.

Programs

Mathematica

Do[ k=n*23^2; f=PowerMod[ 3, k-1, k ] - PowerMod[ 2, k-1, k ]; If[ IntegerQ[ f/k ], Print[ n ] ], {n,1,1000000} ]