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

%I A025003 #18 Sep 19 2020 09:00:18
%S A025003 2,4,8,14,22,33,48,66,90,120,156,202,256,322,400,494,604,734,888,1067,
%T A025003 1272,1512,1790,2107,2472,2890,3364,3903,4515,5207,5990,6875,7868,
%U A025003 8984,10238,11637,13207,14959,16909,19075,21483,24173,27149,30436,34080,38103
%N A025003 a(1) = 2; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1.
%C A025003 Index of first occurrence of n in A090532.
%C A025003 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
%F A025003 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
%e A025003 From _Ya-Ping Lu_, Sep 07 2020: (Start)
%e A025003 a(1) = 2 because f(2) = 2 - pi(2) = 1 and m(2) = 1;
%e A025003 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;
%e A025003 a(2) = 4 because f^2(4) = f(2) = 1 and m(4) = 2;
%e A025003 a(3) = 8 because f^3(8) = f^2(4) = 1 and m(8) = 3. (End)
%p A025003 N:= 50: # to get a(0)..a(N)
%p A025003 V:= Array(0..N):
%p A025003 V[0]:= 1: V[1]:= 2:
%p A025003 m:= 2: p:= 3: g:= 1: n:= 1:
%p A025003 do
%p A025003   if g+p-m-1 >= V[n] then
%p A025003     m:= V[n]+m-g;
%p A025003     n:= n+1;
%p A025003     V[n]:= m;
%p A025003     if n = N then break fi;
%p A025003     g:= V[n-1];
%p A025003   else
%p A025003     g:= g+p-m;
%p A025003     m:= p+1;
%p A025003     p:= nextprime(m);
%p A025003   fi;
%p A025003 od;
%p A025003 convert(V, list); # _Robert Israel_, Sep 08 2020
%o A025003 (Python)
%o A025003 from sympy import prime, primepi
%o A025003 n_last = 0
%o A025003 pi_last = 0
%o A025003 ct_max = -1
%o A025003 for n in range(1, 100001):
%o A025003     ct = 0
%o A025003     pi = pi_last + primepi(n) - primepi(n_last)
%o A025003     n_c = n
%o A025003     pi_c = pi
%o A025003     while n_c > 1:
%o A025003         nc -= pi_c
%o A025003         ct += 1
%o A025003         pi_c -= primepi(n_c + pi_c) - primepi(n_c)
%o A025003     if ct > ct_max:
%o A025003         print(n)
%o A025003         ct_max = ct
%o A025003     n_last = n
%o A025003     pi_last = pi # _Ya-Ping Lu_, Sep 07 2020
%Y A025003 Cf. A000040, A000720, A014688, A014689, A062298, A090532, A332086, A337334.
%K A025003 nonn
%O A025003 1,1
%A A025003 _David W. Wilson_