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A282192
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Number of ways of writing n as a sum of a prime and a squarefree semiprime.
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4
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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
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OFFSET
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0,14
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COMMENTS
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Conjecture: a(n) > 0 for all n > 30.
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LINKS
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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
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EXAMPLE
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a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].
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MAPLE
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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
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MATHEMATICA
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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]
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CROSSREFS
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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
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KEYWORD
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nonn,look
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AUTHOR
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Ilya Gutkovskiy, Feb 15 2017
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STATUS
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approved
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