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
Keywords
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.
Programs
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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 *) -
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
Formula
G.f.: Sum_{i,j>=1} x^(i*j)/((1-x^i)*(1-x^j)). - Seiichi Manyama, Jan 08 2022
Extensions
More terms from Eric Rowland, May 26 2018
Comments