A213250 Numbers n such that the coefficient of x^n in the expansion of Product_{k>=1} (1-x^k)^2 is zero.

7, 11, 12, 17, 18, 21, 22, 25, 32, 37, 39, 41, 42, 43, 46, 47, 49, 54, 57, 58, 60, 62, 65, 67, 68, 72, 74, 75, 76, 81, 82, 87, 88, 90, 92, 95, 97, 98, 99, 106, 107, 109, 111, 112, 113, 116, 117, 120, 122, 123, 125, 126, 128, 130, 132, 136, 137

Offset: 1

Author

William J. Keith, Jun 07 2012

Keywords

• easy
• nonn

Comments

Indices of zero entries in A002107.

Asymptotic density is 1.

Contains A093519, numbers with no representation as sum of two or fewer pentagonal numbers.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002107, A093519.

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), this sequence (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322043 (m=15).

Programs

Julia

# DedekindEta is defined in A000594.
function A213250List(upto)
    eta = DedekindEta(upto, 2)
    [n - 1 for (n, z) in enumerate(eta) if z == 0] end
println(A213250List(140))  # Peter Luschny, Jul 19 2022

Mathematica

LongPoly = Series[Product[1 - q^n, {n, 1, 300}]^2, {q, 0, 300}]; ZeroTable = {}; For[i = 1, i < 301, i++, If[Coefficient[LongPoly, q^i] == 0, AppendTo[ZeroTable, i]]]; ZeroTable

PARI

x='x+O('x^200);
v=Vec(eta(x)^2 - 1);
for(k=1,#v,if(v[k]==0,print1(k,", ")));
/* Joerg Arndt, Jun 07 2012 */