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

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, 60, 63, 8, 66, 69, 72, 75, 78, 13, 81, 84, 87, 90, 93, 96, 99, 102, 16, 105, 108, 19, 111, 114, 11, 117, 120, 14, 123, 126, 22, 129, 132, 135, 138, 141, 25, 144, 147, 150, 153, 156, 28

Offset: 0

Author

Paul D. Hanna, Apr 25 2003; revised May 01 2005

Keywords

• nonn

Comments

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.

Examples

			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, ...}.
A083350(x)^3 = A(x) = 1 + 3x + 6x^2 + 4x^3 + 9x^4 + 12x^5 + 7x^6 + ...
		

Links

• 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.

Crossrefs

Cf. A106213, A106214, A083350, A106216.

Programs

Mathematica

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]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 66}] (* Jean-François Alcover, Jul 25 2018, translated from PARI *)

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))}