1,1

This sequence is equivalent to all of the following sets (written in increasing order):

- 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;

- all integers with an odd number of representations as x^2 + 2y^2 for odd x and y; and

- elements of A192628 which are congruent to 3 (mod 8).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Joshua N. Cooper, Dennis Eichhorn and Kevin O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, Vol. 2, No. 4 (2006), pp. 499-522; arXiv preprint, arXiv:math/0506496 [math.NT], 2005.

Joshua N. Cooper and Alexander W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq., Vol. 16 (2013), Article #13.1.8; alternative link.

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.

6 is not in the sequence: since it is squarefree, k must be 1, but 6 cannot be written as p^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.

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.

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 *)

(SageMath)

prec = 2^10

L = []

for n in range(1, prec, 2):

n = Integer(n)

sfp = n.squarefree_part()

if mod(sfp, 8) == 3 and sfp.is_prime() and mod(n.ord(sfp), 4) == 1:

L.append(n)

print(L)

(SageMath)

def BPS(n): #binary power series

return sum([q^s for s in n])

prec = 2^14

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

q = R.gen()

dList = [(2*n+1)^2 for n in range(0, (sqrt(prec)-1)/2)]

dSeries = BPS(dList)

print((dSeries^3).exponents()[:128])

Cf. A192628.

Sequence in context: A219527 A161429 A079544 * A163183 A007520 A294912

Adjacent sequences: A192714 A192715 A192716 * A192718 A192719 A192720

Alexander Riasanovsky, Dec 31 2012

approved