cp's OEIS Frontend

A192718 Elements of A192628 which are congruent to 7 (mod 8) (equivalently, 7 (mod 16)).

%I A192718 #31 Mar 07 2020 12:25:19
%S A192718 7,55,71,87,103,119,183,263,279,343,375,391,439,455,519,551,567,583,
%T A192718 615,631,647,695,711,727,759,775,791,823,855,871,887,903,951,967,1015,
%U A192718 1047,1079,1095,1111,1127,1159,1175,1191,1223,1239,1271,1303,1319,1367
%N A192718 Elements of A192628 which are congruent to 7 (mod 8) (equivalently, 7 (mod 16)).
%C A192718 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.
%C A192718 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.
%C A192718 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.
%D A192718 J. Cooper and A. Riasanovsky, On the reciprocal of the binary generating function for the sum-of-divisors, Journal of Integer Sequences (accepted).
%D A192718 J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.
%H A192718 J. N. Cooper and A. W. N. Riasanovsky, <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, 2012; <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cooper/cooper3.html">J. Int. Seq. 16 (2013) #13.1.8</a>
%o A192718 (Sage)
%o A192718 prec = 2^12
%o A192718 R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
%o A192718 q = R.gen()
%o A192718 sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())
%o A192718 SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])
%o A192718 SigmaBarSeries = 1/SigmaSeries
%o A192718 SigmaBarList = SigmaBarSeries.exponents()
%o A192718 SigmaBar7Mod8 = [m for m in SigmaBarList if mod(m, 8) == 7]
%o A192718 print(SigmaBar7Mod8)
%K A192718 nonn
%O A192718 1,1
%A A192718 _Alexander Riasanovsky_, Dec 31 2012