A058360 Number of partitions of n whose reciprocal sum is an integer.

1, 1, 1, 2, 2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 19, 23, 25, 31, 33, 38, 42, 51, 57, 66, 75, 86, 97, 109, 122, 138, 155, 177, 200, 230, 253, 287, 320, 363, 405, 456, 507, 572, 639, 707, 785, 877, 971, 1079, 1198, 1334, 1476, 1635, 1802, 2002, 2213, 2445, 2700

Offset: 1

Author

Robert G. Wilson v, Jan 25 2002, Sep 30 2009

Keywords

• nonn

Comments

Also the number of ways to express an integer as the sum of unit fractions such that the sum of the denominators is n.

Examples

			a(12) = 7 because the partitions of 12 whose reciprocal sum is an integer are: {{6, 3, 2, 1}, {4, 4, 2, 1, 1}, {3, 3, 3, 1, 1, 1}, {2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}. Individually their reciprocal sums are: 2, 3, 4, 3, 6, 9 and 12.
		

References

• From a question posted to the news group comp.soft-sys.math.mathematica by "Juan" (erfa11(AT)hotmail.com) at Steven M. Christensen and Associates, Inc. and MathTensor, Inc. Jan 22, 2002 08:46:57 +0000 (UTC).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..80

Crossrefs

Cf. A066824, A051908.

Programs

Mathematica

(* first do *) << "Combinatorica`"; (* then *) f[n_] := Block[{c = i = 0, k = PartitionsP@n, p = {n}}, While[i < k, If[ IntegerQ[ Plus @@ (1/p)], c++ ]; i++; p = NextPartition@ p]; c]; Array[f, 61]
Table[Count[IntegerPartitions[n],?(IntegerQ[Total[1/#]]&)],{n,70}] (* _Harvey P. Dale, Sep 10 2022 *)

PARI

a(n)=my(s); forpart(v=n,if(type(sum(i=1,#v,1/v[i]))=="t_INT",s++)); s \\ Charles R Greathouse IV, Dec 15 2020

PARI

b(n)=if(n<4,return(n==0)); my(s); forpart(v=n, if(type(sum(i=1, #v, 1/v[i]))=="t_INT", s++), [2,n]); s
a(n)=my(s=1); vector(n,i,s+=b(i)) \\ Charles R Greathouse IV, Dec 15 2020