A348665 Number of partitions of n into 3 parts whose smallest and middle parts divide n.

0, 0, 1, 1, 1, 3, 1, 3, 3, 3, 1, 10, 1, 3, 6, 6, 1, 10, 1, 10, 6, 3, 1, 21, 3, 3, 6, 10, 1, 21, 1, 10, 6, 3, 6, 28, 1, 3, 6, 21, 1, 21, 1, 10, 15, 3, 1, 36, 3, 10, 6, 10, 1, 21, 6, 21, 6, 3, 1, 55, 1, 3, 15, 15, 6, 21, 1, 10, 6, 21, 1, 55, 1, 3, 15, 10, 6, 21, 1, 36, 10, 3, 1, 55

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(n/j) * c(n/i), where c(n) = 1 - ceiling(n) + floor(n).

Example

a(6) = 3; The partitions of 6 into 3 parts are [1,1,4], [1,2,3] and [2,2,2]. For each of the 3 partitions, both the smallest part and the middle part divide 6, so a(6) = 3.

Mathematica

c[n_] := 1 - Ceiling[n] + Floor[n]; a[n_] := Sum[c[n/j] * c[n/i], {j, 1, Floor[n/3]}, {i, j, Floor[(n - j)/2]}]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)

Prog

(PARI) A348665(n) = { my(ds=divisors(n)); sum(i=1, #ds, sum(j=1, i, ((n-(ds[i]+ds[j]))>=ds[i]))); }; \\ Antti Karttunen, Dec 14 2021

Crossrefs

Sequence in context: A275367 A023136 A337333 * A152774 A275820 A210146

Adjacent sequences: A348662 A348663 A348664 * A348666 A348667 A348668

Keyword

nonn

Author

Wesley Ivan Hurt, Oct 30 2021

Status

approved