A192628 Nonvanishing exponents of the reciprocal of the modulo 2 generating function for the sum-of-divisor function.

0, 1, 3, 7, 9, 11, 19, 25, 43, 49, 55, 59, 67, 71, 75, 81, 83, 87, 99, 103, 107, 119, 121, 131, 139, 147, 163, 169, 171, 179, 183, 211, 225, 227, 243, 251, 263, 275, 279, 283, 289, 307, 331, 343, 347, 361, 363, 375, 379, 387, 391, 419, 439, 441, 443, 455

Offset: 0

Author

Alexander Riasanovsky, Dec 31 2012

Keywords

• nonn

Comments

Consider sigma, the sum-of-positive-divisor function with s(0) := 1. Let Sigma(q) be the *binary* generating function for sigma, namely

Sigma(q) := sigma(0)q^0 + sigma(1)q^1 + sigma(2)q^2 + sigma(3)q^3 + sigma(4)q^4 + ...

More precisely, we require that Sigma(q) is binary in the sense of reducing all coefficients modulo 2. Thus, the coefficient of q^k is 0 if sigma(k) is even, odd otherwise. One could equivalently define Sigma(q) to be the sum of all q^k (for k nonnegative) such that sigma(k) is odd. The terms of the given sequence are the exponents of the nonvanishing monomials of the reciprocal 1/Sigma(q). Other equivalent definitions for this sequence can be discovered through appeals to representation theory.

Density upper bound: 1/16. Conjectured density: 1/32. Contains only 0 and positive integers congruent to 1 and 3 (mod 8) and 7 (mod 16).

Congruence class:

*0 (mod 8): 0, density 0

*1 (mod 8): odd squares, density 0

*3 (mod 8): integers of the form (p^e)(k^2) for p prime congruent to 3 (mod 8), e congruent to 1 (mod 4), and k odd and coprime to p, density 0

*7 (mod 16): conjectured density 1/32 with upper bound 1/16.

After a(0)=0, these are the positive integers which have an odd number of representations as a sum of positive integers which have odd divisor sum. A positive integer k has odd divisor sum if and only if k is a square or twice a square (A028982). For example, a(2) = 3 can be represented as: 2+1, 1+2, or 1+1+1, 3 representations

References

• J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.

Links

• J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8

Crossrefs

Cf. A028982.

Programs

Sage

prec = 2^14
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
def sigma(n):
    return sum(Integer(n).divisors())
def BPS(n): #binary power series
    return sum([q^s for s in n])
sigmaList = [0] + [n for n in range(1, prec) if mod(sigma(n), 2) == 1]
sigmaSeries = BPS(sigmaList)
print((1/sigmaSeries).exponents()[:128])