A237712 a(n) = |{0 < k < n: k*n + pi(k*n) is prime}|, where pi(.) is given by A000720.

0, 1, 1, 1, 0, 1, 3, 1, 3, 1, 2, 3, 4, 3, 3, 2, 2, 4, 4, 1, 5, 2, 2, 4, 2, 6, 8, 5, 6, 3, 4, 5, 2, 4, 3, 3, 8, 5, 8, 6, 4, 3, 10, 6, 6, 5, 1, 7, 4, 4, 6, 9, 6, 9, 5, 4, 6, 10, 3, 7, 7, 6, 3, 8, 13, 5, 8, 3, 9, 11, 4, 8, 6, 8, 11, 11, 11, 12, 13, 12, 10, 6, 7, 7, 4, 16, 10, 8, 9, 4, 6, 14, 11, 7, 4, 13, 10, 13, 8, 10

Offset

1,7

Comments

Conjecture: a(n) > 0 for all n > 5.

This implies that there are infinitely many positive integers m with m + pi(m) prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500

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

Example

a(6) = 1 since 2*6 + pi(2*6) = 12 + 5 = 17 is prime.
a(47) = 1 since 21*47 + pi(21*47) = 987 + 166 = 1153 is prime.

Mathematica

p[n_]:=PrimeQ[n+PrimePi[n]]
a[n_]:=Sum[If[p[k*n], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A000720, A237578.

Sequence in context: A353415 A078929 A030728 * A227920 A138291 A201681

Adjacent sequences: A237709 A237710 A237711 * A237713 A237714 A237715

Keyword

nonn

Author

Zhi-Wei Sun, Feb 24 2014

Status

approved