A302481 Number of partitions of n into two parts with the smaller part prime and the larger part nonprime.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 4, 3, 6, 4, 6, 3, 5, 4, 5, 3, 7, 3, 7, 6, 7, 5, 8, 4, 7, 5, 7, 5, 9, 4, 8, 5, 9, 6, 9, 4, 8, 6, 9, 6, 10, 4, 9, 8, 10, 6, 11, 5, 11, 9, 10, 6, 11, 5, 10, 7, 11, 7, 12, 5, 12, 8, 11, 8

Offset

1,11

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor(n/2)} c(i) * (1 - c(n-i)), where c is the prime characteristic (A010051).

Example

a(11) = 3; 11 = 9+2 = 8+3 = 6+5, smaller parts are prime, larger nonprime.

Mathematica

Table[Sum[(1 - (PrimePi[n - i] - PrimePi[n - i - 1])) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[n/2]}], {n, 100}]
Table[Count[Boole[PrimeQ[#]&/@IntegerPartitions[n, {2}]], _?(#=={0, 1}&)], {n, 90}] (* Harvey P. Dale, Jan 05 2020 *)

Prog

(PARI) a(n) = sum(i=1, n\2, isprime(i)*(1-isprime(n-i))); \\ Michel Marcus, Apr 09 2018

Crossrefs

Cf. A010051, A302480.

Sequence in context: A085034 A119323 A102299 * A306542 A335171 A259652

Adjacent sequences: A302478 A302479 A302480 * A302482 A302483 A302484

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 08 2018

Status

approved