A307343 Number of partitions of n into 3 mutually distinct, mutually nonadjacent prime parts.

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

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