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

1,13

Comments

Alkan, Booker, & Luca prove that every nonnegative integer appears infinitely often.

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.

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000720, A335337 (when k appears), A334714 (partial sums).

Sequence in context: A319394 A278347 A120936 * A214438 A173432 A101675

Adjacent sequences: A335291 A335292 A335293 * A335295 A335296 A335297

Keyword

nonn,easy

Author

Altug Alkan, Jun 01 2020

Status

approved