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 A307343 Number of partitions of n into 3 mutually distinct, mutually nonadjacent prime parts. 2
 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, 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, 6, 11, 1, 11, 4, 11, 6, 12, 3, 16, 4, 12, 6, 14, 4, 18 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,26 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 Index entries for sequences related to partitions FORMULA 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. EXAMPLE a(18) = 1; 18 = 2 + 5 + 11, which is the only partition of 18 into 3 mutually nonadjacent prime parts. MAPLE 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); # second Maple program: b:= proc(n, i) option remember; `if`(n=0, [1, 0\$3], `if`(i<1, [0\$4], zip((x, y)-> x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0\$3], b(n-ithprime(i), i-2)[1..3])[]], 0))) end: a:= n-> b(n, numtheory[pi](n))[4]: seq(a(n), n=1..200); # Alois P. Heinz, Apr 05 2019 MATHEMATICA 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}] CROSSREFS Cf. A000720, A010051, A125688. Sequence in context: A290253 A097637 A161094 * A340034 A331310 A241597 Adjacent sequences: A307340 A307341 A307342 * A307344 A307345 A307346 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Apr 02 2019 STATUS approved

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Last modified September 19 14:43 EDT 2024. Contains 376013 sequences. (Running on oeis4.)