A192717 - OEIS

%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