A083945 Least integer coefficients of A(x), where 1<=a(n)<=5, such that A(x)^(1/5) consists entirely of integer coefficients.

1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 4, 5, 5, 5, 5, 3, 5, 5, 5, 5, 2, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 2, 5, 5, 5, 5, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 5, 5, 5, 5, 3, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset

0,2

Comments

More generally, "least integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m>0. Is this sequence periodic?

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Mathematica

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/5), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

Crossrefs

Cf. A083952, A083953, A083954, A083946.

Sequence in context: A254609 A133707 A171372 * A125563 A093704 A271509

Adjacent sequences: A083942 A083943 A083944 * A083946 A083947 A083948

Keyword

nonn

Author

Paul D. Hanna, May 09 2003

Extensions

More terms from Robert G. Wilson v, Jul 26 2005

Status

approved