This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A282192 #18 Feb 16 2025 08:33:40
%S A282192 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,
%T A282192 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,
%U A282192 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
%N A282192 Number of ways of writing n as a sum of a prime and a squarefree semiprime.
%C A282192 Conjecture: a(n) > 0 for all n > 30.
%H A282192 Robert Israel, <a href="/A282192/b282192.txt">Table of n, a(n) for n = 0..10000</a>
%H A282192 Ilya Gutkovskiy, <a href="/A282192/a282192.pdf">Extended graphical example</a>
%H A282192 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>
%H A282192 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%e A282192 a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].
%p A282192 N:= 200: # for a(0)..a(N)
%p A282192 P:= select(isprime, [2,seq(i,i=3..N,2)]): nP:= nops(P):
%p A282192 SFS:= NULL: j:= nP:
%p A282192 for i from 1 to nP while j > 0 do
%p A282192 while P[i]*P[j] > N do j:= j-1; if j = 0 then break fi; od:
%p A282192 SFS:= SFS, op(map(`*`,P[1..min(i-1,j)],P[i]))
%p A282192 od:
%p A282192 gS:= add(x^i,i=[SFS]):
%p A282192 gP:= add(x^P[i],i=1..nP):
%p A282192 g:= gP*gS:
%p A282192 [seq(coeff(g,x,i),i=0..N)]; # _Robert Israel_, Jun 15 2020
%t A282192 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]
%Y A282192 Cf. A000040, A006881, A098983, A100949, A258139, A282318, A282355.
%K A282192 nonn,look
%O A282192 0,14
%A A282192 _Ilya Gutkovskiy_, Feb 15 2017