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A302538
Number of partitions of n into two parts (p,q) with p <= q such that p is squarefree and q is semiprime.
2
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, 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, 7, 8, 6, 11, 5, 5, 8, 9, 7, 6, 9, 9, 6, 5, 7, 11, 6
OFFSET
1,7
FORMULA
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.
MAPLE
N:= 100: # for a(1)..a(N)
V:= Vector(N):
SF:= select(numtheory:-issqrfree, [$1..N]): nSF:= nops(SF):
P:= select(isprime, [2, seq(i, i=3..N/2)]):
SP:= sort(select(`<=`, [seq(seq(P[i]*P[j], i=1..j), j=1..nops(P))], N)): nSP:=
nops(SP):
j0:= 1:
for i from 1 to nSF while j0 <= nSP do
x:= SF[i];
while SP[j0] < x do
j0:= j0+1;
if j0 > nSP then break fi;
if SP[j0] + x > N then j0:= nSP+1; break fi;
od;
R:= select(`<=`, x +~ SP[j0 .. nSP], N);
V[R]:= V[R] +~ 1;
od:
convert(V, list); # Robert Israel, Apr 07 2020
MATHEMATICA
Table[Sum[MoebiusMu[i]^2 KroneckerDelta[PrimeOmega[n - i], 2], {i, Floor[n/2]}], {n, 100}]
CROSSREFS
KEYWORD
nonn,easy,look
AUTHOR
Wesley Ivan Hurt, Apr 18 2018
STATUS
approved