A192628 -id:A192628 - cp's OEIS frontend

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

7, 55, 71, 87, 103, 119, 183, 263, 279, 343, 375, 391, 439, 455, 519, 551, 567, 583, 615, 631, 647, 695, 711, 727, 759, 775, 791, 823, 855, 871, 887, 903, 951, 967, 1015, 1047, 1079, 1095, 1111, 1127, 1159, 1175, 1191, 1223, 1239, 1271, 1303, 1319, 1367

Offset: 1

Alexander Riasanovsky, Dec 31 2012

nonn

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.
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.
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.

J. Cooper and A. Riasanovsky, On the reciprocal of the binary generating function for the sum-of-divisors, Journal of Integer Sequences (accepted).
J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.

J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8

prec = 2^12
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())
SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])
SigmaBarSeries = 1/SigmaSeries
SigmaBarList = SigmaBarSeries.exponents()
SigmaBar7Mod8 = [m for m in SigmaBarList if mod(m, 8) == 7]
print(SigmaBar7Mod8)

A210450 Numbers n such that 16n + 7 is in A192628.

Original entry on oeis.org

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, 48, 49, 51, 53, 54, 55, 56, 59, 60, 63, 65, 67, 68, 69, 70, 72, 73, 74, 76, 77, 79, 81, 82, 85, 86, 89, 93, 96, 97, 98, 100, 102, 103, 105, 106, 107, 109, 110

Offset: 1

Alexander Riasanovsky, Jan 20 2013

nonn

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.

J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, arXiv:math/0506496 [math.NT], 2005.
J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, Journal of Integer Sequences, Vol. 16 (2013), #13.1.8.

Cf. A192718, A192628.

prec = 2^12
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())
SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])
SigmaBarSeries = 1/SigmaSeries
SigmaBarList = SigmaBarSeries.exponents()
reduced = [(m-7)/16 for m in SigmaBarList if mod(m, 8) == 7]
print(reduced[:128])

A192717 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.

Original entry on oeis.org

3, 11, 19, 43, 59, 67, 75, 83, 99, 107, 131, 139, 147, 163, 171, 179, 211, 227, 243, 251, 275, 283, 307, 331, 347, 363, 379, 387, 419, 443, 467, 475, 491, 499, 507, 523, 531, 539, 547, 563, 571, 587, 603, 619, 643, 659, 683, 691, 739, 747, 787, 811, 827

Offset: 1

Alexander Riasanovsky, Dec 31 2012

nonn

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).

			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.

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.

Cf. A192628.

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 *)

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)

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])

A210449 Numbers that are the sum of three triangular numbers an odd number of ways.

Original entry on oeis.org

0, 1, 2, 5, 7, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 26, 28, 30, 31, 34, 35, 38, 41, 43, 45, 47, 48, 52, 55, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 73, 75, 77, 80, 82, 85, 86, 92, 93, 98, 101, 103, 107, 108, 110, 111, 113, 116, 118, 120, 121, 127

Offset: 1

Alexander Riasanovsky, Jan 20 2013

nonn

Reduce the elements of A192717 by subtracting 3 and dividing by 8. This makes sense since the elements of A192717 are congruent to 3 (mod 8).

A positive integer n belongs to this sequence precisely when n can be written as t + 2u for triangular numbers t, u an odd number of times, equivalently, written as t + u + v for triangular numbers t, u, v, an odd number of times.

			For n = 0, 1 representation: 0 + 0 + 0; so 0 belongs to this sequence.
For n = 1, 3 representations: 1 + 0 + 0, 0 + 1 + 0, 0 + 0 + 1; so 1 belongs.
For n = 2, 3 representations: 1 + 1 + 0, 1 + 0 + 1, 0 + 1 + 1; so 2 belongs.
For n = 3, 4 representations: 3 + 0 + 0, 0 + 3 + 0, 0 + 0 + 3, 1 + 1 + 1; so 3 does not belong.
For n = 4, 6 representations: 3 + 1 + 0, 3 + 0 + 1, 1 + 3 + 0, 1 + 0 + 3, 0 + 3 + 1, 0 + 1 + 3; so 4 does not belong.
...

J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, arXiv:math/0506496 [math.NT], 2005.
J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012.

Cf. A192717, A192628.

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()
tList = [(n*(n+1))//2 for n in range(0, floor(-1+sqrt(8*prec+1))//2)]
tSeries = BPS(tList)
print((tSeries^3).exponents()[:128])

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

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A210450 Numbers n such that 16n + 7 is in A192628.

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A192717 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.

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A210449 Numbers that are the sum of three triangular numbers an odd number of ways.

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Sage