A337334 -id:A337334 - cp's OEIS frontend

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

David W. Wilson

nonn

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

			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)

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

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

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

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

A334614 a(n) = pi(prime(n) - n) + n, where pi is the prime counting function.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 11, 13, 15, 18, 19, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 45, 47, 48, 50, 51, 53, 55, 57, 60, 61, 65, 66, 67, 68, 70, 72, 74, 76, 77, 79, 81, 82, 85, 88, 89, 91, 93, 94, 95, 99, 101, 102, 104, 105, 106, 107, 108, 112, 116, 117

Offset: 1

Ya-Ping Lu, Sep 08 2020

nonn

It can be shown that a(n) > a(n-1) >= 1 and a(n) <= 2*n - 1 < 2*n (see proofs in the Links section).

Ya-Ping Lu, Proofs of the two observations in the Comments section

Cf. A000040, A000720, A014688, A014689, A062298, A065328, A097933.

Cf. A332086, A337334.

Table[PrimePi[Prime[n] - n] + n, {n, 1, 64}] (* Amiram Eldar, Sep 09 2020 *)

a(n) = n + primepi(prime(n) - n); \\ Michel Marcus, Sep 09 2020

from sympy import prime, primepi
for n in range(1, 100001):
    a_n = primepi(prime(n) - n) + n
    print(a_n)

a(n) = A000720(A014689(n)) + n.

a(n) = A065328(n) + n. - Michel Marcus, Sep 12 2020

cp's OEIS Frontend

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

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