A295630 Number of partitions of n into two distinct parts that are not both prime.

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

Links

Table of n, a(n) for n=1..70.

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

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

Cf. A010051, A295629.

Sequence in context: A077563 A055256 A369985 * A029147 A228571 A224908

Adjacent sequences: A295627 A295628 A295629 * A295631 A295632 A295633

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 24 2017

Status

approved