A083349 -id:A083349 - cp's OEIS frontend

A083350 Integer coefficients of a power series A(x) such that A(x)^3 = A083349(x).

1, 1, 1, -1, 3, 0, -6, 17, -17, -19, 114, -215, 111, 609, -2084, 2947, 1187, -16252, 38872, -32709, -87431, 390618, -673709, 47692, 3018098, -8616766, 9761812, 13605710, -84546525, 171930010, -77194029, -610108400, 2090199824, -2940478260, -1840404119, 19501756943, -46202080484

Offset: 0

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

sign

Self-convolution cube equals A083349.

A083349 is the minimal permutation of the positive integers having a self-convolution cube-root consisting entirely of integers.

			A083349(x)^(1/3) = A(x) = 1 + x + x^2 - x^3 + 3x^4 + 0x^5 - 6x^6 + ...

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.

Cf. A083349, A106213, A106214.

n = 40; 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[]]]]];
CoefficientList[A^(1/3) + O[x]^n, x] (* Jean-François Alcover, Jul 26 2018, from 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))^(1/3),n))}

A106213 Positions n where A083349(n) = 1 (mod 3).

Original entry on oeis.org

0, 3, 6, 24, 33, 42, 45, 54, 60, 66, 69, 78, 87, 93, 102, 108, 111, 120, 129, 144, 153, 162, 165, 174, 183, 186, 195, 204, 213, 216, 225, 228, 231, 237, 240, 258, 261, 267, 279, 285, 300, 306, 312, 321, 333, 336, 351, 363, 366, 384, 390, 408, 414, 417, 420, 429

Offset: 0

Paul D. Hanna, May 01 2005

nonn

A083349 is the minimal permutation of the positive integers having a self-convolution cube-root.

Cf. A083349, A106214, A083350.

PARI

a(n) = 0 (mod 3) for n>=0.

A106214 Positions n where A083349(n) = 2 (mod 3).

Original entry on oeis.org

9, 21, 27, 48, 51, 72, 75, 81, 96, 105, 114, 117, 123, 126, 132, 135, 147, 156, 159, 177, 180, 189, 198, 201, 207, 222, 234, 252, 255, 264, 273, 288, 291, 294, 309, 315, 324, 339, 357, 360, 378, 387, 393, 399, 402, 432, 438, 444, 450, 456

Offset: 0

Paul D. Hanna, May 01 2005

nonn

A083349 is the minimal permutation of the positive integers having a self-convolution cube-root.

Cf. A083349, A106213, A083350.

a(n) = 0 (mod 3) for n>=0.

A110879 Let a_0 = 1 and for n > 0, let a_n be the smallest positive integer not already in the sequence such that (a_0 + a_1 x + a-2x^2 + ....)^(1/3) has integer coefficients. (Hanna's A083349). Let f(n) = n th term in the present sequence. Then a_0 + a_1 x + a_2 x^2 + ... = (1-x)^f(1) (1-x^2)^f(2) (1-x^3)^f(3) ....

Original entry on oeis.org

-1, -2, -3, 5, 1, -3, -3, 7, 6, -7, -23, 15, 12, 28, -48, -25, -10, 165, 4, -274, -408, 927, 932, -1179, -3745, 2906, 7620, -1471, -21283, 1593, 40509, 18877, -93870, -53839, 153551, 204285, -293171, -462306, 307359, 1227141, -282147, -2368041, -1025023, 5041701, 4100247, -7457707, -15096708

Offset: 1

Barry Brent (barrybrent(AT)member.ams.org), Sep 19 2005

sign

The preprint reference asks for a generating function for Hanna's sequence. Terms of present sequence are the exponents in an infinite product for Hanna's sequence. They were obtained from terms of Hanna's sequence with the cited theorem in Apostol and Mobius inversion.

Apostol, T., Introduction to Analytic Number Theory, Springer-Verlag 1976, Theorem 14.8, p. 323.

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.

Cf. A083349.

A083951 Least increasing integer coefficients such that A(x)^(1/3) has only integer coefficients.

Original entry on oeis.org

1, 3, 6, 7, 9, 12, 13, 15, 18, 21, 24, 27, 28, 30, 33, 34, 36, 39, 41, 42, 45, 47, 48, 51, 52, 54, 57, 60, 63, 66, 69, 72, 75, 77, 78, 81, 83, 84, 87, 88, 90, 93, 94, 96, 99, 100, 102, 105, 108, 111, 114, 116, 117, 120, 121, 123, 126, 127, 129, 132, 133, 135, 138, 139

Offset: 0

Paul D. Hanna, May 09 2003

nonn

a(k) == 1 (mod 3) at k=0,3,6,12,15,24,39,42,45,54,57,60,63,66,... a(k) == 2 (mod 3) at k=18,21,33,36,51,...

Robert G. Wilson v, Table of n, a(n) for n = 0..5000.

Cf. A083349, A083953.

a[0] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, s = Sum[ a[i]*x^i, {i, 0, n - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^n)^(1/3), {x, 0, n}], x]]] != True, k++ ]; k]; Array[ a, 70] (* Robert G. Wilson v, Sep 19 2008 *)

Three non-ascending values in the range 77 to 84 replaced with those from the b-file. - R. J. Mathar, Jan 14 2009

cp's OEIS Frontend

A083350 Integer coefficients of a power series A(x) such that A(x)^3 = A083349(x).

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A106213 Positions n where A083349(n) = 1 (mod 3).

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A106214 Positions n where A083349(n) = 2 (mod 3).

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A083951 Least increasing integer coefficients such that A(x)^(1/3) has only integer coefficients.

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