A237497 - OEIS

A237497

a(n) = |{0 < k <= n/2: pi(k*(n-k)) is prime}|, where pi(.) is given by A000720.

0, 0, 0, 2, 2, 1, 2, 1, 1, 0, 1, 4, 3, 1, 1, 1, 3, 2, 6, 2, 2, 2, 4, 1, 1, 3, 3, 3, 1, 3, 3, 7, 4, 5, 4, 6, 5, 5, 3, 3, 3, 5, 7, 4, 1, 6, 7, 7, 5, 4, 1, 2, 3, 5, 5, 6, 8, 8, 6, 4, 9, 8, 6, 3, 7, 9, 6, 5, 4, 10, 5, 4, 6, 6, 4, 9, 10, 6, 8, 7

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OFFSET

1,4

COMMENTS

Conjecture: a(n) > 0 for all n > 10, and a(n) = 1 for no n > 51. Moreover, for any integer n > 10, there is a positive integer k < n with 2*k + 1 and pi(k*(n-k)) both prime.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

EXAMPLE

a(6) = 1 since 6 = 1 + 5 with pi(1*5) = 3 prime.

a(8) = 1 since 8 = 2 + 6 with pi(2*6) = pi(12) = 5 prime.

a(25) = 1 since 25 = 4 + 21 with pi(4*21) = pi(84) = 23 prime.

a(51) = 1 since 51 = 14 + 37 with pi(14*37) = pi(518) = 97 prime.

MATHEMATICA

p[k_, m_]:=PrimeQ[PrimePi[k*m]]

a[n_]:=Sum[If[p[k, n-k], 1, 0], {k, 1, n/2}]

Table[a[n], {n, 1, 80}]