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A192628 Nonvanishing exponents of the reciprocal of the modulo 2 generating function for the sum-of-divisor function. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Table of n, a(n) for n=0..55.

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

PROG

(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]

CROSSREFS

Cf. A028982.

Sequence in context: A197504 A167800 A270834 * A003538 A018596 A191181

Adjacent sequences:  A192625 A192626 A192627 * A192629 A192630 A192631

KEYWORD

nonn

AUTHOR

Alexander Riasanovsky, Dec 31 2012

STATUS

approved

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Last modified July 19 12:35 EDT 2019. Contains 325159 sequences. (Running on oeis4.)