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A192628
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Nonvanishing exponents of the reciprocal of the modulo 2 generating function for the sum-of-divisor function.
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4
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.
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LINKS
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PROG
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(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])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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