A061535 -id:A061535 - cp's OEIS frontend

A337334 a(n) = pi(b(n)), where pi is the prime counting function (A000720) and b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

1, 1, 2, 3, 4, 5, 7, 9, 11, 14, 16, 21, 24, 30, 35, 42, 48, 58, 67, 78, 91, 103, 121, 138, 158, 181, 205, 233, 266, 298, 337, 378, 429, 480, 539, 602, 674, 751, 838, 930, 1031, 1147, 1274, 1402, 1556, 1715, 1896, 2090, 2296, 2527, 2777, 3047, 3340, 3669, 4016

Offset: 0

Ya-Ping Lu, Aug 23 2020

nonn

It can be proved that this is an increasing sequence from the theorem of Lu and Deng (see LINKS), which states "the prime gap of a prime number is less than or equal to the prime count of the prime number”, or prime(n+1) - prime(n) <= pi(prime(n)).

			a(1) = pi(b(1)) = pi(a(0) + b(0)) = pi(1 + 1) = pi(2) = 1
a(2) = pi(b(2)) = pi(a(1) + b(1)) = pi(1 + 2) = pi(3) = 2
a(3) = pi(b(3)) = pi(a(2) + b(2)) = pi(2 + 3) = pi(5) = 3
a(4) = pi(b(4)) = pi(a(3) + b(3)) = pi(3 + 5) = pi(8) = 4
a(54)= pi(b(54))= pi(a(53)+ b(53))= pi(3669+34327)=pi(37996)=4016

Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.

Cf. A000720 (pi), A014688 (prime(n)+n), A332086.

A337334 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        numtheory[pi](A061535(n)) ;
    end if;
end proc:
seq(A337334(n),n=0..20) ; # R. J. Mathar, Jun 18 2021

from sympy import primepi
a_last = 1
b_last = 1
for n in range(1, 1001):
    b = a_last + b_last
    a = primepi(b)
    print(a)
    a_last = a
    b_last = b

a(n) = pi(b(n)), where b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

a(n) = A000720(A061535(n)), n>=1. - R. J. Mathar, Jun 18 2021

a(0) inserted by R. J. Mathar, Jun 18 2021

A340716 Lexicographically earliest sequence of positive integers with as many distinct values as possible such that for any n > 0, a(n + pi(n)) = a(n) (where pi(n) = A000720(n) corresponds to the number of prime numbers <= n).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 2, 3, 5, 4, 2, 3, 5, 6, 4, 2, 7, 3, 5, 6, 4, 8, 2, 7, 9, 3, 5, 6, 4, 10, 8, 2, 7, 9, 3, 5, 11, 6, 4, 12, 10, 8, 2, 7, 9, 3, 13, 5, 11, 6, 4, 14, 12, 10, 15, 8, 2, 7, 9, 16, 3, 13, 5, 11, 6, 4, 17, 14, 12, 10, 15, 8, 2, 18, 7, 9, 19, 16, 3

Offset: 1

Rémy Sigrist, Jan 17 2021

The condition "with as many distinct values as possible" means here that for any distinct m and n, provided the orbits of m and n under the map x -> x + pi(x) do not merge, then a(m) <> a(n).

This sequence has similarities with A003602 (A003602(2*n) = A003602(n)) and with A163491 (A163491(n+ceiling(n/2)) = A163491(n)).

			The first terms, alongside n + pi(n), are:
  n   a(n)  n + pi(n)
  --  ----  ---------
   1     1          1
   2     2          3
   3     2          5
   4     3          6
   5     2          8
   6     3          9
   7     4         11
   8     2         12
   9     3         13
  10     5         14
  11     4         16
  12     2         17

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A003602, A163491 and A340717 for similar sequences.

Cf. A000720, A095117.

u=0; for (n=1, #a=vector(80), if (a[n]==0, a[n]=u++); print1 (a[n]", "); m=n+primepi(n); if (m<=#a, a[m]=a[n]))

a(n) = 2 iff n belongs to A061535.

a(A095116(n)) = n + 1.

cp's OEIS Frontend

A337334 a(n) = pi(b(n)), where pi is the prime counting function (A000720) and b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Formula

Extensions