A253807 Primitive part of A006190(n), n >= 1.

1, 3, 10, 11, 109, 12, 1189, 119, 1297, 131, 141481, 118, 1543321, 1429, 15445, 14159, 183642229, 1299, 2003229469, 14041, 1837837, 170039, 238367471761, 14158, 23854956949, 1854841, 2186871697, 1670761, 309400794703549

Offset: 1

Author

Wolfdieter Lang, Jan 19 2015

Keywords

• nonn
• easy

Comments

A006190(n) = Product_{k divides n} a(k), n >= 1.

Crossrefs

Cf. A000010, A006190, A008683, A013595.

Cf. A061446, A008555, A105603.

Programs

Mathematica

(* b = A006190 *) b[0] = 0; b[1] = 1; b[n_] := b[n] = 3*b[n-1] + b[n-2]; a[n_] := Product[b[d]^MoebiusMu[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 20 2015 *)

Formula

a(n) = ((3-sqrt(13))/2)^phi(n)*cyclotomic(n, -(11 - 3*sqrt13)/2) for n >= 1 and a(1) = 1, where phi is Euler's totient A000010 and the coefficient table for the cyclotomic polynomials is given in A013595.

a(n) = Product_{d|n} A006190(d)^mu(n/d), where mu = A008683, n >= 1.