

A322043


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


8



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
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OFFSET

1,1


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} (1x^m)^15 are given in A010822.


REFERENCES

Van der Blij, F. "The function tau(n) of S. Ramanujan (an expository lecture)." Math. Student 18 (1950): 8399. See page 85.


LINKS

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


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(n1)  3*a(n2) + a(n3) 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


MATHEMATICA

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


PROG

(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


CROSSREFS

Cf. A000594, A010822, A302057.
Sequence in context: A181968 A261537 A142209 * A293089 A177120 A165555
Adjacent sequences: A322040 A322041 A322042 * A322044 A322045 A322046


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Dec 07 2018


EXTENSIONS

a(4)a(7) supplied by Rémy Sigrist, Dec 07 2018, from the bfile for A010822.
a(8)a(19) from Seiichi Manyama, Dec 07 2018
a(20)a(31) from JeanFrançois Alcover, Dec 20 2018
More terms from Joerg Arndt, Dec 20 2018


STATUS

approved



