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A237615 a(n) = |{0 < k < n: k^2 + k - 1 and pi(k*n) are both prime}|, where pi(.) is given by A000720. 5
0, 0, 1, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 4, 1, 3, 4, 4, 2, 4, 3, 6, 2, 2, 2, 3, 7, 4, 3, 4, 5, 6, 1, 3, 2, 3, 9, 3, 3, 4, 7, 5, 8, 5, 2, 2, 5, 5, 4, 5, 6, 4, 5, 6, 10, 6, 6, 10, 9, 9, 10, 12, 2, 8, 7, 3, 6, 6, 4, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

(ii) For each n = 4, 5, ..., there is a positive integer k < n with k^2 + k - 1 and pi(k*n) + 1 both prime. Also, for any integer n > 6, there is a positive integer k < n with k^2 + k - 1 and pi(k*n) - 1 both prime.

(iii) For every integer n > 15, there is a positive integer k < n such that pi(k) - 1 and pi(k*n) are both prime.

Note that part (i) is a refinement of the first assertion in the comments in A237578.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Zhi-Wei Sun, A combinatorial conjecture on primes, a message to Number Theory List, Feb. 9, 2014.

EXAMPLE

a(8) = 1 since 4^2 + 4 - 1 = 19 and pi(4*8) = 11 are both prime.

a(33) = 1 since 28^2 + 28 - 1 = 811 and pi(28*33) = 157 are both prime.

MATHEMATICA

p[k_, n_]:=PrimeQ[k^2+k-1]&&PrimeQ[PrimePi[k*n]]

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

Table[a[n], {n, 1, 70}]

CROSSREFS

Cf. A000040, A000720, A002327, A045546, A237578, A237597, A237598.

Sequence in context: A305432 A305298 A298824 * A256132 A303476 A187201

Adjacent sequences:  A237612 A237613 A237614 * A237616 A237617 A237618

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 10 2014

STATUS

approved

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Last modified March 19 13:08 EDT 2019. Contains 321330 sequences. (Running on oeis4.)