A303206 Number of partitions of n into two prime parts (p,q) such that |q-p| is squarefree.

0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 4, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 0, 5, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 6, 1, 0, 1, 3, 0, 0, 0, 1, 1, 0, 0, 5, 1, 0, 1, 4, 0, 0, 0, 3, 1, 0, 0, 7, 0, 0

Offset

1,16

Links

Table of n, a(n) for n=1..86.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} A010051(i) * A010051(n-i) * A008966(n-2i).

Mathematica

Table[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]) MoebiusMu[n - 2 i]^2, {i, Floor[(n - 1)/2]}], {n, 100}]
Table[Count[IntegerPartitions[n, {2}], _?(AllTrue[#, PrimeQ]&&SquareFreeQ[#[[1]]-#[[2]]]&)], {n, 100}] (* Harvey P. Dale, Aug 05 2023 *)

Prog

(PARI) a(n) = sum(i=1, (n-1)\2, isprime(i)*isprime(n-i)*issquarefree(n-2*i)); \\ Michel Marcus, Apr 21 2018; corrected by Jun 14 2022

Crossrefs

Cf. A008966, A010051.

Sequence in context: A364046 A001899 A059882 * A094247 A053694 A085862

Adjacent sequences: A303203 A303204 A303205 * A303207 A303208 A303209

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 19 2018

Status

approved