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%I #20 Apr 05 2019 18:50:58
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,1,0,2,1,1,1,2,1,2,3,1,
%T 2,2,2,3,4,1,2,2,3,3,4,2,5,2,3,5,7,3,7,2,5,5,9,2,8,3,9,5,10,1,8,4,10,
%U 6,11,1,11,4,11,6,12,3,16,4,12,6,14,4,18
%N Number of partitions of n into 3 mutually distinct, mutually nonadjacent prime parts.
%H Alois P. Heinz, <a href="/A307343/b307343.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_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} A010051(i) * A010051(k) * A010051(n-i-k) * (1-floor((pi(k)+1)/pi(i))) * (1-floor((pi(i)+1)/pi(n-i-k))), where pi is the prime counting function.
%e a(18) = 1; 18 = 2 + 5 + 11, which is the only partition of 18 into 3 mutually nonadjacent prime parts.
%p with(numtheory): A307343:=n->add(add((pi(k)-pi(k-1))*(pi(i)-pi(i-1))*(pi(n-i-k)-pi(n-i-k-1))*(1-floor((pi(k)+1)/pi(i)))*(1-floor((pi(i)+1)/pi(n-i-k))), i=k+1..floor((n-k-1)/2)), k=1..floor((n-1)/3)): seq(A307343(n), n=1..150);
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, [0$4],
%p zip((x, y)-> x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$3],
%p b(n-ithprime(i), i-2)[1..3])[]], 0)))
%p end:
%p a:= n-> b(n, numtheory[pi](n))[4]:
%p seq(a(n), n=1..200); # _Alois P. Heinz_, Apr 05 2019
%t Table[Sum[Sum[(1 - Floor[(PrimePi[k] + 1)/PrimePi[i]]) (1 - Floor[(PrimePi[i] + 1)/PrimePi[n - i - k]]) (PrimePi[i] - PrimePi[i - 1])*(PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]), {i, k + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}]
%Y Cf. A000720, A010051, A125688.
%K nonn,easy
%O 1,26
%A _Wesley Ivan Hurt_, Apr 02 2019
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