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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*(k-1)) 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, ..., n-1.

LINKS

Zhi-Wei 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

Zhi-Wei Sun, Feb 09 2014

STATUS

approved

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Last modified May 25 08:26 EDT 2020. Contains 334585 sequences. (Running on oeis4.)