

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
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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

ZhiWei Sun, Table of n, a(n) for n = 1..5000
ZhiWei 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+k1]&&PrimeQ[PrimePi[k*n]]
a[n_]:=Sum[If[p[k, n], 1, 0], {k, 1, n1}]
Table[a[n], {n, 1, 70}]


CROSSREFS

Cf. A000040, A000720, A002327, A045546, A237578, A237597, A237598.
Sequence in context: A305298 A298824 A343384 * A343190 A256132 A340057
Adjacent sequences: A237612 A237613 A237614 * A237616 A237617 A237618


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 10 2014


STATUS

approved



