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%I #28 Apr 08 2020 01:40:43
%S 0,0,0,0,1,1,2,1,1,1,3,3,1,1,3,4,3,1,1,3,2,2,2,3,3,3,5,6,3,2,3,4,2,2,
%T 5,7,4,4,6,8,6,4,4,6,5,4,5,7,6,4,4,8,5,5,4,8,6,5,6,8,7,6,7,9,8,6,6,12,
%U 7,8,6,11,5,5,8,9,7,6,9,9,6,5,7,11,6
%N Number of partitions of n into two parts (p,q) with p <= q such that p is squarefree and q is semiprime.
%H Robert Israel, <a href="/A302538/b302538.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor((n-1)/2)} mu(i)^2 * [Omega(n-i) = 2], where [] is the Iverson bracket, mu = A008683 and Omega = A001222.
%p N:= 100: # for a(1)..a(N)
%p V:= Vector(N):
%p SF:= select(numtheory:-issqrfree, [$1..N]): nSF:= nops(SF):
%p P:= select(isprime, [2,seq(i,i=3..N/2)]):
%p SP:= sort(select(`<=`,[seq(seq(P[i]*P[j],i=1..j),j=1..nops(P))],N)): nSP:=
%p nops(SP):
%p j0:= 1:
%p for i from 1 to nSF while j0 <= nSP do
%p x:= SF[i];
%p while SP[j0] < x do
%p j0:= j0+1;
%p if j0 > nSP then break fi;
%p if SP[j0] + x > N then j0:= nSP+1; break fi;
%p od;
%p R:= select(`<=`,x +~ SP[j0 .. nSP], N);
%p V[R]:= V[R] +~ 1;
%p od:
%p convert(V,list); # _Robert Israel_, Apr 07 2020
%t Table[Sum[MoebiusMu[i]^2 KroneckerDelta[PrimeOmega[n - i], 2], {i, Floor[n/2]}], {n, 100}]
%Y Cf. A001222, A008683, A303119.
%K nonn,easy,look
%O 1,7
%A _Wesley Ivan Hurt_, Apr 18 2018