This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A083954 #20 Jan 29 2025 12:39:30
%S A083954 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,
%T A083954 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,
%U A083954 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
%N A083954 Least integer coefficients of A(x), where 1<=a(n)<=4, such that A(x)^(1/4) consists entirely of integer coefficients.
%C A083954 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?
%C A083954 From _M. F. Hasler_, Jan 27 2025: (Start)
%C A083954 The sequence does not seem to become periodic.
%C A083954 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.
%C A083954 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.
%C A083954 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)
%H A083954 Robert G. Wilson v, <a href="/A083954/b083954.txt">Table of n, a(n) for n = 0..5000</a>.
%H A083954 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%t A083954 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 *)
%o A083954 (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
%Y A083954 Cf. A083952, A083953, A083945, A083946.
%K A083954 nonn
%O A083954 0,2
%A A083954 _Paul D. Hanna_, May 09 2003
%E A083954 More terms from _Robert G. Wilson v_, Jul 26 2005