A274628 - OEIS

%I #44 Sep 28 2024 20:27:25

%S 1,4,7,13,15,26,25,39,40,54,49,79,63,88,88,112,93,140,109,159,142,170,

%T 143,224,168,216,202,255,199,304,219,308,268,316,274,404,281,370,338,

%U 438,323,484,345,481,433,484,389,611,422,566,492,607,459,684,508,692

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

%C Number of integer solutions to a*b - c*d = n such that a > c >= 0 and b > d >= 0. - _David Radcliffe_, Mar 28 2019

%H Giovanni Resta, <a href="/A274628/b274628.txt">Table of n, a(n) for n = 1..10000</a>

%H Brandon Dong, Soren Dupont, and W. Theo Waitkus, <a href="https://arxiv.org/abs/2409.15480">Raney Transducers and the Lowest Point of the p-Lagrange spectrum</a>, arXiv:2409.15480 [math.NT], 2024. See p. 25.

%H Sandie Han, Ariane M. Masuda, Satyanand Singh and Johann Thiel, <a href="https://doi.org/10.37236/5684">Orphans in Forests of Linear Fractional Transformations</a>, Electronic Journal of Combinatorics, Vol. 23, No. 3 (2016), Article P3.6. See Fig. 4.

%H Melvyn B. Nathanson, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.8.790">Pairs of matrices in GL_2(R_{>0}) that freely generate</a>, Amer. Math. Monthly, Vol. 122 No. 8 (2015), pp. 790-792. See Theorem 7.

%F a(n) = A002133(n) + 2*A000203(n) - A000005(n). - _David Radcliffe_, Mar 28 2019

%F G.f.: Sum_{i,j>=1} x^(i*j)/((1-x^i)*(1-x^j)). - _Seiichi Manyama_, Jan 08 2022

%t 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 *)

%o (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

%o (Python)

%o from sympy import divisor_sigma

%o 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

%Y Cf. A000005, A000203, A002133, A274629.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jul 07 2016

%E More terms from _Eric Rowland_, May 26 2018