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

1,4

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.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A066824, A051908.

Sequence in context: A194815 A029080 A147652 * A241901 A238213 A193942

Adjacent sequences: A058357 A058358 A058359 * A058361 A058362 A058363

Keyword

nonn

Author

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

Status

approved