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A237578 a(n) = |{0 < k < n: pi(k*n) is prime}|, where pi(.) is given by A000720. 31
0, 0, 2, 2, 1, 3, 2, 1, 2, 2, 4, 4, 1, 4, 2, 5, 5, 6, 2, 5, 4, 6, 3, 7, 3, 3, 7, 5, 5, 5, 10, 9, 3, 7, 6, 5, 12, 3, 3, 9, 10, 11, 12, 7, 3, 5, 11, 9, 7, 10, 12, 9, 10, 8, 12, 11, 10, 17, 15, 13, 14, 18, 4, 17, 10, 9, 15, 11, 14, 11, 23, 11, 9, 13, 12, 12, 12, 11, 14, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 5, 8, 13. Moreover, for each n = 1, 2, 3, ..., there is a positive integer k < 3*sqrt(n) + 3 with pi(k*n) prime.

Note that the least positive integer k with pi(k*38) prime is 21 < 3*sqrt(38) + 3 < 21.5.

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(5) = 1 since pi(1*5) = 3 is prime.

a(8) = 1 since pi(4*8) = 11 is prime.

a(13) = 1 since pi(10*13) = pi(130) = 31 is prime.

a(38) = 3 since pi(21*38) = pi(798) = 139, pi(28*38) = pi(1064) = 179 and pi(31*38) = pi(1178) = 193 are all prime.

MATHEMATICA

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

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

CROSSREFS

Cf. A000040, A000720, A237453, A237496, A237497.

Sequence in context: A243926 A281013 A190683 * A026146 A221057 A094366

Adjacent sequences:  A237575 A237576 A237577 * A237579 A237580 A237581

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 09 2014

STATUS

approved

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)