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A274628
Nathanson's orphan-counting function h(n).
3
1, 4, 7, 13, 15, 26, 25, 39, 40, 54, 49, 79, 63, 88, 88, 112, 93, 140, 109, 159, 142, 170, 143, 224, 168, 216, 202, 255, 199, 304, 219, 308, 268, 316, 274, 404, 281, 370, 338, 438, 323, 484, 345, 481, 433, 484, 389, 611, 422, 566, 492, 607, 459, 684, 508, 692
OFFSET
1,2
COMMENTS
Number of integer solutions to a*b - c*d = n such that a > c >= 0 and b > d >= 0. - David Radcliffe, Mar 28 2019
LINKS
Brandon Dong, Soren Dupont, and W. Theo Waitkus, Raney Transducers and the Lowest Point of the p-Lagrange spectrum, arXiv:2409.15480 [math.NT], 2024. See p. 25.
Sandie Han, Ariane M. Masuda, Satyanand Singh and Johann Thiel, Orphans in Forests of Linear Fractional Transformations, Electronic Journal of Combinatorics, Vol. 23, No. 3 (2016), Article P3.6. See Fig. 4.
Melvyn B. Nathanson, Pairs of matrices in GL_2(R_{>0}) that freely generate, Amer. Math. Monthly, Vol. 122 No. 8 (2015), pp. 790-792. See Theorem 7.
FORMULA
a(n) = A002133(n) + 2*A000203(n) - A000005(n). - David Radcliffe, Mar 28 2019
G.f.: Sum_{i,j>=1} x^(i*j)/((1-x^i)*(1-x^j)). - Seiichi Manyama, Jan 08 2022
MATHEMATICA
Table[Total[Function[parts, Count[CountDistinct /@ IntegerPartitions[n, All, parts], 2]] /@ Subsets[Range[n], {2}]] + 2 DivisorSigma[1, n] - DivisorSigma[0, n], {n, 1, 100}] (* Eric Rowland, May 26 2018 *)
PROG
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(i=1, N, sum(j=1, N\i, x^(i*j)/((1-x^i)*(1-x^j))))) \\ Seiichi Manyama, Jan 08 2022
(Python)
from sympy import divisor_sigma
def A274628(n): return int(sum(divisor_sigma(j, 0)*divisor_sigma(n-j, 0) for j in range(1, (n-1>>1)+1)) + ((divisor_sigma(n+1>>1, 0)**2 if n-1&1 else 0)-divisor_sigma(n, 0)+3*divisor_sigma(n)>>1)) # Chai Wah Wu, Aug 30 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 07 2016
EXTENSIONS
More terms from Eric Rowland, May 26 2018
STATUS
approved