A316898 Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 4, 1, 3, 1, 1, 1, 3, 1, 8, 3, 1, 1, 9, 2, 11, 3, 3, 3, 5, 2, 7, 6, 4, 7, 12, 5, 14, 6, 11, 12, 25, 11, 27, 17, 15, 19, 25, 9, 37, 20, 21, 19, 31, 19, 38, 33, 26, 37, 38, 36, 64, 39, 46, 53, 63, 39, 80, 63, 65, 66, 94, 59, 105

Offset

1,22

Comments

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

Records: 1, 2, 4, 8, 9, 11, 12, 14, 25, 27, 37, 38, 64, 80, 94, 105, 119, 154, 184, ..., . - Robert G. Wilson v, Jul 18 2018

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..150

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Example

The a(37) = 8 partitions: (20,12,5), (15,12,10), (24,8,3,2), (15,10,6,6), (20,5,4,4,4), (15,10,6,3,3), (14,7,7,7,2), (10,10,10,5,2).

Mathematica

f[n_] := Block[{k = 1, s = 0, lmt = 1 + PartitionsP@ n}, While[k < lmt, s += Length[ Select[ IntegerPartitions[n, {k, k}], GCD @@ # == 1 && IntegerQ[1/Sum[1/m, {m, #}]] &]]; k++]; s]; Array[f, 50] (* slightly modified by Robert G. Wilson v, Jul 17 2018 *) (* or *)
ric[n_, p_, s_] := If[n==0, If[IntegerQ[1/s] && GCD @@ p == 1, c++], Do[ If[s + 1/i <= 1, ric[n-i, Append[p, i], s + 1/i]], {i, Min[p[[-1]], n], 1, -1}]]; a[n_] := (c=0; Do[ric[n-j, {j}, 1/j], {j, n}]; c); Array[a, 50] (* Giovanni Resta, Jul 18 2018 *)

Crossrefs

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888, A316889, A316890, A316891, A316892, A316893, A316904.

Sequence in context: A133827 A176192 A028633 * A351254 A356303 A238429

Adjacent sequences: A316895 A316896 A316897 * A316899 A316900 A316901

Keyword

nonn

Author

Gus Wiseman, Jul 16 2018

Extensions

a(51)-a(91) from Robert G. Wilson v, Jul 17 2018

Status

approved