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

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}]

Crossrefs

Cf. A000040, A000720, A237284, A237291, A237453, A237496.

Sequence in context: A221649 A090406 A152723 * A281191 A324496 A137454

Adjacent sequences: A237494 A237495 A237496 * A237498 A237499 A237500

Keyword

nonn

Author

Zhi-Wei Sun, Feb 08 2014

Status

approved