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A238890 a(n) = |{0 < k <= n: prime(k*n) - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x. 2
1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 6, 1, 1, 4, 4, 1, 5, 3, 5, 5, 4, 5, 1, 2, 5, 7, 6, 5, 2, 2, 4, 4, 4, 10, 6, 5, 5, 4, 6, 8, 7, 5, 8, 5, 8, 5, 3, 5, 9, 6, 7, 2, 2, 4, 6, 7, 8, 11, 8, 8, 10, 6, 8, 10, 2, 5, 11, 7, 5, 10, 10, 8, 7, 9, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 28.

(ii) If n > 7 is not equal to 34, then prime(k*n) + pi(k*n) is prime for some k = 1, ..., n.

The conjecture implies that there are infinitely many primes p with p - pi(pi(p)) (or p + pi(pi(p)) prime.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

EXAMPLE

a(5) = 1 since prime(3*5) - pi(3*5) = 47 - 6 = 41 is prime.

a(28) = 1 since prime(18*28) - pi(18*28) = prime(504) - pi(504) = 3607 - 96 = 3511 is prime.

MATHEMATICA

p[k_]:=PrimeQ[Prime[k]-PrimePi[k]]

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

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

CROSSREFS

Cf. A000040, A000720, A237578, A237712, A238573, A238576, A238878, A238881.

Sequence in context: A320107 A190321 A338409 * A266968 A237593 A338169

Adjacent sequences:  A238887 A238888 A238889 * A238891 A238892 A238893

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 06 2014

STATUS

approved

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Last modified April 16 03:59 EDT 2021. Contains 343030 sequences. (Running on oeis4.)