A274628 Nathanson's orphan-counting function h(n).

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

Giovanni Resta, Table of n, a(n) for n = 1..10000

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

Cf. A000005, A000203, A002133, A274629.

Sequence in context: A310798 A310799 A310800 * A060751 A230745 A191201

Adjacent sequences: A274625 A274626 A274627 * A274629 A274630 A274631

Keyword

nonn

Author

N. J. A. Sloane, Jul 07 2016

Extensions

More terms from Eric Rowland, May 26 2018

Status

approved