A348531 Number of partitions of n into 3 parts where at least one of the parts divides the product of the other two.

0, 0, 1, 1, 2, 3, 4, 5, 7, 7, 10, 10, 14, 14, 17, 17, 22, 20, 28, 25, 29, 30, 38, 32, 43, 40, 45, 43, 57, 45, 62, 56, 62, 63, 70, 61, 84, 74, 81, 74, 98, 78, 108, 92, 95, 102, 120, 95, 127, 109, 123, 116, 149, 118, 142, 129, 145, 147, 173, 126, 182, 163, 164, 164, 184, 158, 211

Offset

1,5

Links

Table of n, a(n) for n=1..67.

Index entries for sequences related to partitions

Formula

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

Example

a(9) = 7; All of the partitions of 9 (into 3 such parts) satisfy these conditions. They are (1,1,7), (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4) and (3,3,3).
a(10) = 7; The partitions of 10 into 3 such parts are (1,1,8), (1,2,7), (1,3,6), (1,4,5), (2,2,6), (2,4,4) and (3,3,4).

Mathematica

Block[{c}, c[n_] := 1 - Ceiling[n] + Floor[n]; Array[Sum[Sum[Sign[c[i*(# - i - j)/j] + c[j*(# - i - j)/i] + c[i*j/(# - i - j)]], {i, j, Floor[(# - j)/2]}], {j, Floor[#/3]} ] &, 67]] (* Michael De Vlieger, Oct 21 2021 *)

Crossrefs

Sequence in context: A342542 A338671 A343246 * A237824 A227972 A266620

Adjacent sequences: A348528 A348529 A348530 * A348532 A348533 A348534

Keyword

nonn

Author

Wesley Ivan Hurt, Oct 21 2021

Status

approved