A282192 Number of ways of writing n as a sum of a prime and a squarefree semiprime.

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

Offset: 0

Author

Ilya Gutkovskiy, Feb 15 2017

Keywords

• nonn
• look

Comments

Conjecture: a(n) > 0 for all n > 30.

Examples

			a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Semiprime
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Cf. A000040, A006881, A098983, A100949, A258139, A282318, A282355.

Programs

Maple

N:= 200: # for a(0)..a(N)
P:= select(isprime, [2,seq(i,i=3..N,2)]): nP:= nops(P):
SFS:= NULL: j:= nP:
for i from 1 to nP while j > 0 do
  while P[i]*P[j] > N do j:= j-1; if j = 0 then break fi; od:
  SFS:= SFS, op(map(`*`,P[1..min(i-1,j)],P[i]))
od:
gS:= add(x^i,i=[SFS]):
gP:= add(x^P[i],i=1..nP):
g:= gP*gS:
[seq(coeff(g,x,i),i=0..N)]; # Robert Israel, Jun 15 2020

Mathematica

nmax = 108; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}] Sum[MoebiusMu[k]^2 Floor[2/PrimeOmega[k]] Floor[PrimeOmega[k]/2] x^k, {k, 2, nmax}], {x, 0, nmax}], x]