

A237595


a(n) = {1 <= k <= n: n + pi(k^2) is prime}, where pi(.) is given by A000720.


3



0, 1, 3, 0, 3, 1, 3, 3, 3, 1, 5, 2, 6, 3, 4, 2, 6, 3, 7, 3, 2, 6, 8, 1, 10, 3, 5, 8, 9, 2, 9, 6, 3, 5, 14, 5, 11, 6, 9, 3, 13, 8, 11, 8, 8, 6, 8, 8, 11, 9, 6, 12, 15, 10, 11, 5, 11, 12, 13, 9, 12, 9, 5, 17, 15, 9, 18, 13, 11, 12
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 4.
(ii) If n > 1 then n + pi(k*(k1)) is prime for some k = 1, ..., n.
(iii) For any integer n > 1, there is a positive integer k <= (n+1)/2 such that pi(n + k*(k+1)/2) is prime.
(iv) Any integer n > 1 can be written as p + pi(k*(k+1)/2), where p is a prime and k is among 1, ..., n1.


LINKS

ZhiWei 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(2) = 1 since 2 + pi(1^2) = 2 is prime.
a(6) = 1 since 6 + pi(6^2) = 6 + 11 = 17 is prime.
a(10) = 1 since 10 + pi(5^2) = 10 + 9 = 19 is prime.
a(21) = 2 since 21 + pi(2^2) = 23 and 21 + pi(9^2) = 43 are both prime.
a(24) = 1 since 24 + pi(21^2) = 24 + 85 = 109 is prime.


MATHEMATICA

a[n_]:=Sum[If[PrimeQ[n+PrimePi[k^2]], 1, 0], {k, 1, n}]
Table[a[n], {n, 1, 70}]


CROSSREFS

Cf. A000040, A000720, A237453, A237496, A237497, A237578, A237582.
Sequence in context: A307807 A318504 A318505 * A322575 A166445 A298082
Adjacent sequences: A237592 A237593 A237594 * A237596 A237597 A237598


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 09 2014


STATUS

approved



