cp's OEIS Frontend

A083349 Least positive integers not appearing previously such that the self-convolution cube-root of this sequence consists entirely of integers.

%I A083349 #10 Jul 25 2018 11:19:19
%S A083349 1,3,6,4,9,12,7,15,18,2,21,24,27,30,33,36,39,42,45,48,51,5,54,57,10,
%T A083349 60,63,8,66,69,72,75,78,13,81,84,87,90,93,96,99,102,16,105,108,19,111,
%U A083349 114,11,117,120,14,123,126,22,129,132,135,138,141,25,144,147,150,153,156,28
%N A083349 Least positive integers not appearing previously such that the self-convolution cube-root of this sequence consists entirely of integers.
%C A083349 A permutation of the positive integers. Positive integers congruent to 1 (mod 3) appear in ascending order at positions given by A106213. Positive integers congruent to 2 (mod 3) appear in ascending order at positions given by A106214. The self-convolution cube-root is A083350.
%H A083349 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%e A083349 The self-convolution cube of A083350 equals this sequence: {1, 1, 1, -1, 3, 0, -6, 17, -17, -19, 114, ...}^3 = {1, 3, 6, 4, 9, 12, 7, 15, 18, ...}.
%e A083349 A083350(x)^3 = A(x) = 1 + 3x + 6x^2 + 4x^3 + 9x^4 + 12x^5 + 7x^6 + ...
%t A083349 a[n_] := a[n] = Module[{A, P, t}, A = 1+3x; P = Table[0, 3(n+1)]; P[[1]] = 1; P[[3]] = 2; For[j = 2, j <= n, j++, For[k = 2, k <= 3(n+1), k++, If[P[[k]] == 0, t = Coefficient[(A + k x^j + x^2 O[x]^j)^(1/3), x, j]; If[Denominator[t] == 1, P[[k]] = j+1; A = A + k*x^j; Break[]]]]]; Coefficient[A + x O[x]^n, x, n]];
%t A083349 Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 66}] (* _Jean-François Alcover_, Jul 25 2018, translated from PARI *)
%o A083349 (PARI) {a(n)=local(A=1+3*x,P=vector(3*(n+1)));P[1]=1;P[3]=2; for(j=2,n, for(k=2,3*(n+1),if(P[k]==0, t=polcoeff((A+k*x^j+x^2*O(x^j))^(1/3),j); if(denominator(t)==1,P[k]=j+1;A=A+k*x^j;break)))); return(polcoeff(A+x*O(x^n),n))}
%Y A083349 Cf. A106213, A106214, A083350, A106216.
%K A083349 nonn
%O A083349 0,2
%A A083349 _Paul D. Hanna_, Apr 25 2003; revised May 01 2005