A335294 a(n) = pi(n) - pi(Sum_{k=1..n-1} a(k)) with a(1) = 1, where pi() is the prime counting function A000720.
1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1
Offset: 1
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- Altug Alkan, Andrew R. Booker, and Florian Luca, On a recursively defined sequence involving the prime counting function, arXiv:2006.08013 [math.NT], 2020.
Programs
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Maple
A[1]:= 1: S:= 1: for n from 2 to 100 do A[n]:= numtheory:-pi(n) - numtheory:-pi(S); S:= S + A[n]; od: seq(A[n],n=1..100); # Robert Israel, Jun 01 2020
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Mathematica
a[1] = 1; a[n_] := a[n] = PrimePi[n] - PrimePi[Sum[a[k], {k, 1, n-1}]]; Array[a, 100] (* Amiram Eldar, Jun 01 2020 *) upto[nn_] := Reap[Block[{i=0, is=0, sp=2, p=2, s=1}, Sow@ 1; Do[ If[n == p, i++; p = NextPrime@ p]; Sow[i - is]; s += i - is; While[ s >= sp, is++; sp = NextPrime@ sp], {n, 2, nn}]]] [[2, 1]]; upto[97] (* Giovanni Resta, Jun 02 2020 *) -
PARI
a=vector(10^2); a[1] = 1; for(n=2, #a, a[n] = primepi(n) - primepi(sum(k=1, n-1, a[k]))); a
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PARI
first(n) = {my(res = vector(n), pp = 0, s = 1, ps=0); primepivec = vector(n); forprime(p = 2, n, primepivec[p] = 1; ); for(i = 2, n, primepivec[i] += primepivec[i-1] ); res[1] = 1; for(i = 2, n, if(isprime(i), pp++); res[i] = pp - ps; s+=(pp-ps); ps = primepivec[s]; ); res } \\ David A. Corneth, Jun 01 2020 -
Python
from sympy import primepi A = [1] S = 1 for n in range(1, 101): A += [primepi(n+1) - primepi(S), ] S += A[n] print(A) # Indranil Ghosh, Jun 21 2020, after Maple
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