OFFSET
1,1
COMMENTS
This is the subsequence/subset of A192628 which contains elements congruent to 7 modulo 8. Equivalently, these elements are also congruent to 7 modulo 16.
By partitioning A192628 into congruence classes k modulo 8, it turns out that it contains only elements congruent to 0, 1, 3, and 7 modulo 8. Further, the congruence classes 0, 1, and 3 modulo 8 are vanishing--having a density asymptotic to 0.
However, the 7 modulo 8 congruence classes appears to have nonzero density, conjectured 1/32. A current upper bound on its density (thus the entire density of A192628) is 1/16.
REFERENCES
J. Cooper and A. Riasanovsky, On the reciprocal of the binary generating function for the sum-of-divisors, Journal of Integer Sequences (accepted).
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
PROG
(Sage)
prec = 2^12
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())
SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])
SigmaBarSeries = 1/SigmaSeries
SigmaBarList = SigmaBarSeries.exponents()
SigmaBar7Mod8 = [m for m in SigmaBarList if mod(m, 8) == 7]
print(SigmaBar7Mod8)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander Riasanovsky, Dec 31 2012
STATUS
approved