A337788 - OEIS

A337788

The number of primes between n exclusive and n+primepi(n) inclusive.

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 5, 5, 4, 4, 4, 4, 4, 4, 5, 5, 4, 4, 4, 5, 4, 4, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 6, 6

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OFFSET

1,9

COMMENTS

There is at least one prime number in the range of (n, n + primepi(n)], or a(n) >= 1, for n >= 2 (see Corollary 1 in the paper by Ya-Ping Lu attached in the links).

See also the Panaitopol link. - Charles R Greathouse IV, Jul 12 2024

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Ya-Ping Lu, Lower Bounds for the Number of Primes in Some Integer Intervals

Laurențiu Panaitopol, Intervals containing prime numbers, NNTDM 7 (2001), 4, pp. 111-114.

FORMULA

a(n) = primepi(n + primepi(n)) - primepi(n)

a(n) = A000720(n + A000720(n)) - A000720(n)

a(n) = A000720(A095117(n)) - A000720(n)

MATHEMATICA

Table[Count[Range[n+1, n+PrimePi[n]], _?PrimeQ], {n, 90}] (* Harvey P. Dale, Aug 28 2024 *)

PROG