A336404 a(n) = floor(n*frac(prime(n)/pi(n))), where frac denotes the fractional part.

0, 1, 2, 3, 2, 1, 6, 6, 2, 2, 4, 10, 2, 12, 13, 7, 12, 7, 17, 2, 19, 5, 21, 19, 5, 12, 24, 26, 9, 16, 29, 15, 21, 19, 26, 3, 22, 35, 16, 31, 38, 27, 34, 3, 9, 3, 41, 6, 13, 27, 48, 3, 37, 3, 24, 46, 54, 17, 31, 44, 17, 3, 17

Offset

2,3

Comments

Conjecture: For any m >= 0 there is a k such that a(k) = m. Also for any reals x and epsilon such that 0 < x < 1 and epsilon > 0, there is a k such that abs(x - frac(prime(k)/pi(k))) < epsilon.

Links

Table of n, a(n) for n=2..64.

Andres Cicuttin, Graph of first 100000 terms

Andres Cicuttin, Graphs of the fractional part of prime(n)/pi(n) and its first differences of the first 2^16 terms

Mathematica

a[n_] := Floor[n*FractionalPart[Prime[n]/PrimePi[n]]]; Table[a[n], {n, 2, 2^6}]

Prog

(PARI) a(n) = floor(n*frac(prime(n)/primepi(n))); \\ Michel Marcus, Jul 21 2020
(PARI) first(n) = {my(t = 2, res = vector(n), pit = 0); forprime(p = 3, oo, if(isprime(t), pit++); res[t-1] = floor(t * frac(p/pit)); t++; if(t-1 > n, return(res)))} \\ David A. Corneth, Aug 22 2020

Crossrefs

Cf. A111114 (floor(prime(n)/pi(n))), A004648 (prime(n) mod n).

Sequence in context: A113908 A355624 A065369 * A370182 A167772 A077870

Adjacent sequences: A336401 A336402 A336403 * A336405 A336406 A336407

Keyword

nonn

Author

Andres Cicuttin, Jul 20 2020

Status

approved