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A282192 Number of ways of writing n as a sum of a prime and a squarefree semiprime. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

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

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

EXAMPLE

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

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]

CROSSREFS

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

Sequence in context: A144740 A283272 A292166 * A049501 A102564 A215703

Adjacent sequences:  A282189 A282190 A282191 * A282193 A282194 A282195

KEYWORD

nonn,look

AUTHOR

Ilya Gutkovskiy, Feb 15 2017

STATUS

approved

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Last modified September 23 22:30 EDT 2020. Contains 337315 sequences. (Running on oeis4.)