A083954 Least integer coefficients of A(x), where 1<=a(n)<=4, such that A(x)^(1/4) consists entirely of integer coefficients.

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

Offset: 0

Author

Paul D. Hanna, May 09 2003

Keywords

• nonn

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?

From M. F. Hasler, Jan 27 2025: (Start)

The sequence does not seem to become periodic.

Positions of '1's are: (0, 8, 60, 64, 72, 96, 100, 112, 116, 148, 160, 176, 184, 200, 240, 248, 268, 288, 304, 328, 336, 360, 376, 380, 384, 400, 408, 420, 424, 448, 460, 472, ...). All seem to be multiples of 4, mostly multiples of 8.

Positions of '3's are: (4, 12, 16, 28, 36, 40, 76, 84, 104, 124, 136, 172, 192, 196, 208, 212, 220, 232, 252, 260, 284, 296, 312, 364, 368, 392, 404, 428, 432, 436, 452, 456, 468, 488, 492, ...). All seem to be (mostly odd) multiples of 4.

The proportions of '1's, '2's, '3's and '4's among the terms are approximately: 6.5%, 18%, 6.5%, 69%. (Roughly the same values for the first 500 or 5000 terms.) (End)

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..5000.
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Crossrefs

Cf. A083952, A083953, A083945, A083946.

Programs

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)^(1/4), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v, Jul 26 2005 *)

PARI

A083954_upto(N=99)=vector(N+1, n, if(n>1, for(k=1,4, denominator(polcoeff(sqrtn(O(x^n)+N+=x^(n-1), 4), n-1))>1|| [n=k, break]); n, N=1)) \\ _M. F. Hasler, Jan 27 2025

Extensions

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