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%I #51 Mar 07 2021 03:20:15
%S 3,11,19,43,59,67,75,83,99,107,131,139,147,163,171,179,211,227,243,
%T 251,275,283,307,331,347,363,379,387,419,443,467,475,491,499,507,523,
%U 531,539,547,563,571,587,603,619,643,659,683,691,739,747,787,811,827
%N Positive integers of the form (p^e)(k^2) for p prime congruent to 3 (mod 8), e congruent to 1 (mod 4), and k an odd integer coprime to p.
%C This sequence is equivalent to all of the following sets (written in increasing order):
%C - all integers the form (p^e)(k^2) for p prime congruent to 3 (mod 8), e congruent to 1 (mod 4), and k an odd number coprime to p;
%C - all integers with an odd number of representations as x^2 + 2y^2 for odd x and y; and
%C - elements of A192628 which are congruent to 3 (mod 8).
%H Amiram Eldar, <a href="/A192717/b192717.txt">Table of n, a(n) for n = 1..10000</a>
%H Joshua N. Cooper, Dennis Eichhorn and Kevin O'Bryant, <a href="https://doi.org/10.1142/S1793042106000693">Reciprocals of binary power series</a>, International Journal of Number Theory, Vol. 2, No. 4 (2006), pp. 499-522; <a href="https://arxiv.org/abs/math/0506496">arXiv preprint</a>, arXiv:math/0506496 [math.NT], 2005.
%H Joshua N. Cooper and Alexander 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>, J. Int. Seq., Vol. 16 (2013), Article #13.1.8; <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">alternative link</a>.
%e 3 is in the sequence since 3 = (3^1)(1^2); 3 is prime and congruent to 3 (mod 8), 1 is congruent to 1 (mod 4), and 1 is an odd integer coprime to 3.
%e 6 is not in the sequence: since it is squarefree, k must be 1, but 6 cannot be written as p^e.
%e 27 is not in the sequence: the only possible values for k are 1 and 3. In the k=1 case, 27 = (3^3)(1^2) does not work since e = 3 is not congruent to 1 (mod 4), and in the k=3 case, 27 = (3^1)(3^2), k=3 and p=3 are not coprime.
%e 243 is in the sequence since 243 = (3^5)(1^2); 3 is prime and congruent to 3 (mod 8), 5 is congruent to 1 (mod 4), and 1 is an odd integer coprime to 3.
%t ofTheFormQ[n_] := If[Length[fin = FactorInteger[n]] == 1 && Mod[fin[[1, 1]], 8] == 3 && Mod[fin[[1, 2]], 4] == 1, True, pe = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ fin); k = Sqrt[n/pe]; fip = FactorInteger[pe]; Length[fip] == 1 && Mod[p = fip[[1, 1]], 8] == 3 && Mod[e = fip[[1, 2]], 4] == 1 && OddQ[k] && CoprimeQ[k, p]]; Select[Range[1, 999, 2], ofTheFormQ] (* _Jean-François Alcover_, Jan 22 2013 *)
%o (Sage)
%o prec = 2^10
%o L = []
%o for n in range(1, prec, 2):
%o n = Integer(n)
%o sfp = n.squarefree_part()
%o if mod(sfp, 8) == 3 and sfp.is_prime() and mod(n.ord(sfp), 4) == 1:
%o L.append(n)
%o print(L)
%o (Sage)
%o def BPS(n): #binary power series
%o return sum([q^s for s in n])
%o prec = 2^14
%o R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
%o q = R.gen()
%o dList = [(2*n+1)^2 for n in range(0, (sqrt(prec)-1)/2)]
%o dSeries = BPS(dList)
%o print((dSeries^3).exponents()[:128])
%Y Cf. A192628.
%K nonn
%O 1,1
%A _Alexander Riasanovsky_, Dec 31 2012
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