A302643 Number of partitions of n into two parts such that the positive difference of the parts is a squarefree semiprime.

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

Offset

1,12

Links

Table of n, a(n) for n=1..81.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} A280710(n-2i).

Example

As 6 is a semiprime, we know that a(6 + 2*k) > 0 for k > 0.

Mathematica

Table[Sum[MoebiusMu[n - 2 i]^2*KroneckerDelta[PrimeOmega[n - 2 i], 2], {i, Floor[(n - 1)/2]}], {n, 100}]

Prog

(PARI) a(n) = sum(i=1, (n-1)\2, bigomega(n-2*i)==2*issquarefree(n-2*i)); \\ Michel Marcus, Apr 11 2018
(PARI) upto(n) = {my(semiprimes = List(), res = vector(n)); forprime(p = 2, sqrtint(n), forprime(q = p+1, n \ p, listput(semiprimes, p * q))); for(i = 1, #semiprimes, forstep(j = semiprimes[i] + 2, n, 2, res[j]++)); res} \\ David A. Corneth, Apr 11 2018

Crossrefs

Cf. A280710, A302604, A302642.

Sequence in context: A164960 A124137 A164092 * A319973 A025804 A042961

Adjacent sequences: A302640 A302641 A302642 * A302644 A302645 A302646

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 10 2018

Status

approved