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A345039
Number of partitions of n into two composite parts that share a nontrivial divisor.
0
0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 1, 3, 0, 4, 0, 4, 2, 4, 0, 6, 1, 5, 3, 6, 0, 8, 0, 7, 4, 7, 3, 10, 0, 8, 5, 10, 0, 12, 0, 10, 8, 10, 0, 14, 2, 13, 7, 12, 0, 16, 5, 14, 8, 13, 0, 19, 0, 14, 11, 15, 6, 20, 0, 16, 10, 20, 0, 22, 0, 17, 15, 18, 6, 24, 0, 22, 12, 19, 0, 27
OFFSET
1,12
FORMULA
a(n) = Sum_{k=1..floor(n/2)} (1 - floor(1/gcd(k,n-k))) * c(k) * c(n-k), where c(n) is the characteristic function of composite numbers.
EXAMPLE
a(12) = 2; (8,4) and (6,6).
a(15) = 1; (9,6).
a(16) = 3; (12,4), (10,6), and (8,8).
a(18) = 4; (14,4), (12,6), (10,8), and (9,9).
MATHEMATICA
Table[Sum[(1 - PrimePi[k] + PrimePi[k - 1]) (1 - PrimePi[n - k] + PrimePi[n - k - 1]) (1 - Floor[1/GCD[k, n - k]]), {k, Floor[n/2]}], {n, 100}]
CROSSREFS
Sequence in context: A354856 A055136 A074397 * A082023 A349438 A078152
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 06 2021
STATUS
approved