login
A295630
Number of partitions of n into two distinct parts that are not both prime.
0
0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 5, 4, 5, 5, 6, 5, 8, 6, 8, 7, 9, 8, 11, 8, 11, 10, 13, 11, 14, 11, 14, 13, 15, 13, 17, 13, 18, 17, 18, 16, 20, 16, 20, 18, 21, 19, 23, 18, 23, 20, 25, 22, 26, 21, 26, 24, 28, 25, 29, 23, 29, 28, 30, 26, 32, 26, 33, 31, 33, 29
OFFSET
1,6
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} 1 - A010051(i)*A010051(n-i).
EXAMPLE
a(12) = 4; The partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). Of these partitions, the parts in (11,1), (10,2), (9,3) and (8,4) are not both prime, so a(12) = 4.
MATHEMATICA
Table[Sum[1 - (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}], {n, 80}]
Table[Count[IntegerPartitions[n, {2}], _?(#[[1]]!=#[[2]]&&Total[Boole[ PrimeQ[ #]]]<2&)], {n, 70}] (* Harvey P. Dale, May 18 2021 *)
CROSSREFS
Sequence in context: A077563 A055256 A369985 * A029147 A228571 A224908
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Nov 24 2017
STATUS
approved