A338521 The number of primes between n-primepi(n) and n.

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

Offset: 1

Author

Ya-Ping Lu, Nov 01 2020

Keywords

• nonn

Comments

There is at least one prime number between n-primepi(n) and n, or a(n) >= 1, for n >= 3 (see Corollary 3 in the paper by Ya_Ping Lu attached in the links).

Links

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

Crossrefs

Cf. A000720, A062298, A337788.

Programs

Mathematica

Array[Subtract @@ Map[PrimePi, {#1 - 1, #1 - #2}] & @@ {#, PrimePi[#]} &, 105] (* Michael De Vlieger, Nov 05 2020 *)

PARI

a(n) = primepi(n - 1) - primepi(n - primepi(n)); \\ Michel Marcus, Nov 01 2020

Python

from sympy import primepi
for n in range(1, 101):
    pi = primepi(n)
    pi_1 = primepi(n - 1)
    a = pi_1 - primepi(n - pi)
    print(a)

Formula

a(n) = primepi(n - 1) - primepi(n - primepi(n)).

a(n) = A000720(n - 1) - A000720(n - A000720(n)).

a(n) = A000720(n -1) - A000720(A062298(n)).