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A322043 Numbers k such that the coefficient of x^k in the expansion of Product_{m >= 1} (1-x^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 (list; graph; refs; listen; history; text; internal format)
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} (1-x^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): 83-99. 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(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

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 *)

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 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

STATUS

approved

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Last modified February 20 14:03 EST 2020. Contains 332078 sequences. (Running on oeis4.)