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A237497 a(n) = |{0 < k <= n/2: pi(k*(n-k)) is prime}|, where pi(.) is given by A000720. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)