A210450 - OEIS

%I #28 Mar 14 2020 07:03:11

%S 0,3,4,5,6,7,11,16,17,21,23,24,27,28,32,34,35,36,38,39,40,43,44,45,47,

%T 48,49,51,53,54,55,56,59,60,63,65,67,68,69,70,72,73,74,76,77,79,81,82,

%U 85,86,89,93,96,97,98,100,102,103,105,106,107,109,110

%N Numbers n such that 16n + 7 is in A192628.

%C Reduce the elements of A192718 (which are the elements of A192628 congruent to 7 (mod 16)) by subtracting 7 and dividing by 16.  In "On the reciprocal of the binary generating function for the sum of divisors", this sequence is precisely the set T.

%H J. N. Cooper, D. Eichhorn, and K. O'Bryant, <a href="https://arxiv.org/abs/math/0506496">Reciprocals of binary power series</a>, arXiv:math/0506496 [math.NT], 2005.

%H J. N. Cooper, D. Eichhorn, and K. O'Bryant, <a href="https://doi.org/10.1142/S1793042106000693">Reciprocals of binary power series</a>, International Journal of Number Theory, 2 no. 4 (2006), 499-522.

%H J. N. Cooper and A. W. N. Riasanovsky, <a href="http://people.math.sc.edu/cooper/Sigma.pdf">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, 2012.

%H J. N. Cooper and A. W. N. Riasanovsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cooper/cooper3.html">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.1.8.

%o (Sage)

%o prec = 2^12

%o R = PowerSeriesRing(GF(2), 'q', default_prec = prec)

%o q = R.gen()

%o sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())

%o SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])

%o SigmaBarSeries = 1/SigmaSeries

%o SigmaBarList = SigmaBarSeries.exponents()

%o reduced = [(m-7)/16 for m in SigmaBarList if mod(m, 8) == 7]

%o print(reduced[:128])

%Y Cf. A192718, A192628.

%K nonn

%O 1,2

%A _Alexander Riasanovsky_, Jan 20 2013