A322043 Numbers k such that the coefficient of x^k in the expansion of Product_{m >= 1} (1-x^m)^15 is zero.

53, 482, 1340, 2627, 4343, 6488, 9062, 12065, 15497, 19358, 23648, 28367, 33515, 39092, 45098, 51533, 58397, 65690, 73412, 81563, 90143, 99152, 108590, 118457, 128753, 139478, 150632, 162215, 174227, 186668, 199538, 212837, 226565, 240722, 255308, 270323, 285767, 301640, 317942, 334673, 351833, 369422, 387440

Offset: 1

Author

N. J. A. Sloane, Dec 07 2018

Keywords

• nonn

Comments

Van der Blij, discussing the conjecture that the Ramanujan numbers tau(k) (see A000594) are never zero, mentions that a certain "Ferguson" had shown that 52 is a member of the current sequence. No details were given, and the 52 appears to be a typo for 53.

The coefficients of the expansion of Product_{m >= 1} (1-x^m)^15 are given in A010822.

References

• F. Van der Blij, "The function tau(n) of S. Ramanujan (an expository lecture)." Math. Student 18 (1950): 83-99. See page 85.

Links

• Joerg Arndt, Table of n, a(n) for n = 1..216

Crossrefs

Cf. A000594, A010822, A302057.

Programs

Mathematica

sigma[k_] := sigma[k] = DivisorSigma[1, k];
a[0] = 1; a[n_] := a[n] = -15/n Sum[sigma[k] a[n-k], {k, 1, n}];
Reap[For[k = 1, k <= 200000, k++, If[a[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 20 2018 *)

PARI

/* start with sufficient memory, e.g., gp -s16G */
x='x+O('x^1000000); v=Vec(eta(x)^15 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ Joerg Arndt, Dec 20 2018

Formula

Conjectures from Colin Barker, Dec 07 2018: (Start)

G.f.: x*(53 + 323*x + 53*x^2) / (1 - x)^3.

a(n) = (429*n^2 - 429*n + 106) / 2.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.

(End)

E.g.f.: (1/2)*exp(x)*(106 + 858*x + 429*x^2). - conjectured by Stefano Spezia, Dec 07 2018 after the conjectures of Colin Barker

Extensions

a(4)-a(7) supplied by Rémy Sigrist, Dec 07 2018, from the b-file for A010822.

a(8)-a(19) from Seiichi Manyama, Dec 07 2018

a(20)-a(31) from Jean-François Alcover, Dec 20 2018

More terms from Joerg Arndt, Dec 20 2018