A303337 Number of rectangles with semiprime area and dimensions (p) X (p+q) such that n = p+q, p < q.

0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 3, 0, 3, 1, 1, 0, 4, 0, 4, 0, 1, 1, 5, 0, 1, 1, 0, 0, 6, 0, 6, 0, 1, 1, 1, 0, 7, 1, 1, 0, 8, 0, 8, 0, 0, 1, 9, 0, 1, 0, 1, 0, 9, 0, 1, 0, 1, 1, 10, 0, 10, 1, 0, 0, 1, 0, 11, 0, 1, 0, 11, 0, 11, 1, 0, 0, 1, 0, 12, 0, 0, 1, 13, 0

Offset

1,7

Comments

From Robert Israel, Jul 30 2020: (Start)

If n is prime, a(n) = A000720(floor(n/2)).

If n is a semiprime, a(n) = 1.

Otherwise a(n) = 0. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} [Omega(n*i) = 2], where [] is the Iverson bracket and Omega = A001222.

Maple

f:= proc(n) if isprime(n) then numtheory:-pi(floor(n/2)) elif numtheory:-bigomega(n)=2 then 1 else 0 fi end proc:
map(f, [$1..100]); # Robert Israel, Jul 30 2020

Mathematica

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

Prog

(PARI) a(n) = sum(i=1, (n-1)\2, bigomega(n*i) == 2); \\ Michel Marcus, Apr 22 2018

Crossrefs

Cf. A000720, A001222, A303336.

Sequence in context: A339846 A029353 A064922 * A340502 A055186 A217760

Adjacent sequences: A303334 A303335 A303336 * A303338 A303339 A303340

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 21 2018

Status

approved