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

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, 17, 17, 18, 19, 19, 20, 21, 22, 22, 22, 23, 23, 24, 25, 25, 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 33, 34, 35, 35, 35, 36, 36, 37, 38, 39, 39, 39, 40, 41, 42, 42, 42, 43, 44, 44

Offset: 1

Author

Altug Alkan, Jul 02 2018

Keywords

• nonn

Comments

This sequence hits every positive integer.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Altug Alkan, A plot of a(n)/n
• Altug Alkan and Orhan Ozgur Aybar, On a Family of Sequences Related to Prime Counting Function, International Journal of Statistics and Probability Vol. 7, No. 6; 2018.
• Rémy Sigrist, C++ program for A316434

Crossrefs

Cf. A000720, A062298.

Programs

Maple

f:= proc(n) option remember: local p;
     p:= numtheory:-pi(n);
     procname(p) + procname(n-p)
end proc:
f(1):= 1: f(2):= 1:
map(f, [$1..100]); # Robert Israel, Jul 03 2018

Mathematica

a[1]=a[2]=1; a[n_] := a[n] = a[PrimePi[n]] + a[n - PrimePi[n]]; Array[a, 75] (* Giovanni Resta, Nov 02 2018 *)

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
(C++) See Links section.

Python

from sympy import primepi
def A316434(n):
    pp = primepi(n)
    return 1 if n == 1 or n == 2 else A316434(pp) + A316434(n-pp) # Chai Wah Wu, Nov 02 2018

Formula

a(n) = a(A000720(n)) + a(A062298(n)) with a(1) = a(2) = 1.

a(n+1) - a(n) = 0 or 1 for all n >= 1.