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A348531
Number of partitions of n into 3 parts where at least one of the parts divides the product of the other two.
1
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Oct 21 2021
STATUS
approved