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Number of partitions of n into two parts such that the positive difference of the parts is a squarefree semiprime.
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%I #9 Apr 16 2018 19:01:50

%S 0,0,0,0,0,0,0,1,0,1,0,2,0,2,0,3,1,3,1,3,1,3,2,4,2,4,2,5,2,5,2,5,2,5,

%T 3,6,4,6,4,7,5,7,5,7,5,7,5,8,5,8,5,8,6,8,6,8,7,8,8,9,8,9,8,10,8,10,9,

%U 10,9,10,10,10,10,10,10,11,10,11,11,11,11

%N Number of partitions of n into two parts such that the positive difference of the parts is a squarefree semiprime.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=1..floor((n-1)/2)} A280710(n-2i).

%e As 6 is a semiprime, we know that a(6 + 2*k) > 0 for k > 0.

%t Table[Sum[MoebiusMu[n - 2 i]^2*KroneckerDelta[PrimeOmega[n - 2 i], 2], {i, Floor[(n - 1)/2]}], {n, 100}]

%o (PARI) a(n) = sum(i=1, (n-1)\2, bigomega(n-2*i)==2*issquarefree(n-2*i)); \\ _Michel Marcus_, Apr 11 2018

%o (PARI) upto(n) = {my(semiprimes = List(), res = vector(n)); forprime(p = 2, sqrtint(n), forprime(q = p+1, n \ p, listput(semiprimes, p * q))); for(i = 1, #semiprimes, forstep(j = semiprimes[i] + 2, n, 2, res[j]++)); res} \\ _David A. Corneth_, Apr 11 2018

%Y Cf. A280710, A302604, A302642.

%K nonn,easy

%O 1,12

%A _Wesley Ivan Hurt_, Apr 10 2018