A025003 a(1) = 2; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1.

2, 4, 8, 14, 22, 33, 48, 66, 90, 120, 156, 202, 256, 322, 400, 494, 604, 734, 888, 1067, 1272, 1512, 1790, 2107, 2472, 2890, 3364, 3903, 4515, 5207, 5990, 6875, 7868, 8984, 10238, 11637, 13207, 14959, 16909, 19075, 21483, 24173, 27149, 30436, 34080, 38103

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Index of first occurrence of n in A090532.

Let b(n) (n >= 0) be the smallest integer k >= 1 that takes n steps to reach 1 iterating the map f: k -> k - pi(k). The sequence {b(n), n >= 0} begins 1, 2, 4, 8, 14, 22, 33, 48, 66, 90, 120, 156, ... and agrees with the present sequence except for b(0). - Ya-Ping Lu, Sep 07 2020

Examples

			From _Ya-Ping Lu_, Sep 07 2020: (Start)
a(1) = 2 because f(2) = 2 - pi(2) = 1 and m(2) = 1;
For the integer 3, since f(3) = 1. m(3) = 1, which is not bigger than m(1) or m(2). So, 3 is not a term in the sequence;
a(2) = 4 because f^2(4) = f(2) = 1 and m(4) = 2;
a(3) = 8 because f^3(8) = f^2(4) = 1 and m(8) = 3. (End)
		

Crossrefs

Cf. A000040, A000720, A014688, A014689, A062298, A090532, A332086, A337334.

Programs

Maple

N:= 50: # to get a(0)..a(N)
V:= Array(0..N):
V[0]:= 1: V[1]:= 2:
m:= 2: p:= 3: g:= 1: n:= 1:
do
  if g+p-m-1 >= V[n] then
    m:= V[n]+m-g;
    n:= n+1;
    V[n]:= m;
    if n = N then break fi;
    g:= V[n-1];
  else
    g:= g+p-m;
    m:= p+1;
    p:= nextprime(m);
  fi;
od;
convert(V, list); # Robert Israel, Sep 08 2020

Python

from sympy import prime, primepi
n_last = 0
pi_last = 0
ct_max = -1
for n in range(1, 100001):
    ct = 0
    pi = pi_last + primepi(n) - primepi(n_last)
    n_c = n
    pi_c = pi
    while n_c > 1:
        nc -= pi_c
        ct += 1
        pi_c -= primepi(n_c + pi_c) - primepi(n_c)
    if ct > ct_max:
        print(n)
        ct_max = ct
    n_last = n
    pi_last = pi # Ya-Ping Lu, Sep 07 2020

Formula

a(n) = min(k: f^n(k) = 1), where f = A062298 and n-fold iteration of f is denoted by f^n. - Ya-Ping Lu, Sep 07 2020