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A316434 a(n) = a(pi(n)) + a(n-pi(n)) with a(1) = a(2) = 1.

%I A316434 #29 Nov 02 2018 14:58:00
%S A316434 1,1,2,2,3,4,4,4,5,6,7,7,8,8,9,10,10,11,11,11,12,12,13,14,15,15,16,16,
%T A316434 17,17,18,19,19,20,21,22,22,22,23,23,24,25,25,25,26,27,28,28,29,30,31,
%U A316434 31,32,32,33,33,34,35,35,35,36,36,37,38,39,39,39,40,41,42,42,42,43,44,44
%N A316434 a(n) = a(pi(n)) + a(n-pi(n)) with a(1) = a(2) = 1.
%C A316434 This sequence hits every positive integer.
%H A316434 Robert Israel, <a href="/A316434/b316434.txt">Table of n, a(n) for n = 1..10000</a>
%H A316434 Altug Alkan, <a href="/A316434/a316434.png">A plot of a(n)/n</a>
%H A316434 Altug Alkan and Orhan Ozgur Aybar, <a href="https://doi.org/10.5539/ijsp.v7n6p149"> On a Family of Sequences Related to Prime Counting Function</a>, International Journal of Statistics and Probability Vol. 7, No. 6; 2018.
%H A316434 Rémy Sigrist, <a href="/A316434/a316434.txt">C++ program for A316434</a>
%F A316434 a(n) = a(A000720(n)) + a(A062298(n)) with a(1) = a(2) = 1.
%F A316434 a(n+1) - a(n) = 0 or 1 for all n >= 1.
%p A316434 f:= proc(n) option remember: local p;
%p A316434      p:= numtheory:-pi(n);
%p A316434      procname(p) + procname(n-p)
%p A316434 end proc:
%p A316434 f(1):= 1: f(2):= 1:
%p A316434 map(f, [$1..100]); # _Robert Israel_, Jul 03 2018
%t A316434 a[1]=a[2]=1; a[n_] := a[n] = a[PrimePi[n]] + a[n - PrimePi[n]]; Array[a, 75] (* _Giovanni Resta_, Nov 02 2018 *)
%o A316434 (PARI) q=vector(75); for(n=1, 2, q[n] = 1); for(n=3, #q, q[n] = q[primepi(n)] + q[n-primepi(n)]); q
%o A316434 (C++) See Links section.
%o A316434 (Python)
%o A316434 from sympy import primepi
%o A316434 def A316434(n):
%o A316434     pp = primepi(n)
%o A316434     return 1 if n == 1 or n == 2 else A316434(pp) + A316434(n-pp) # _Chai Wah Wu_, Nov 02 2018
%Y A316434 Cf. A000720, A062298.
%K A316434 nonn
%O A316434 1,3
%A A316434 _Altug Alkan_, Jul 02 2018