A112045 Positions of primes (A000040) among nonsquares A000037.

1, 2, 3, 5, 8, 10, 13, 15, 19, 24, 26, 31, 35, 37, 41, 46, 52, 54, 59, 63, 65, 71, 74, 80, 88, 91, 93, 97, 99, 103, 116, 120, 126, 128, 137, 139, 145, 151, 155, 160, 166, 168, 178, 180, 183, 185, 197, 209, 212, 214, 218, 224, 226, 236, 241, 247, 253, 255, 261

Offset: 1

Author

Antti Karttunen, Aug 27 2005

Keywords

• nonn

Comments

Also: Distance of prime(n) from the integer part of its square root. - M. F. Hasler, Oct 19 2018

Crossrefs

Cf. A000037, A000040, A028391, A071403 (the original name describes this sequence).

Programs

Mathematica

f[n_]:=n-IntegerPart[Sqrt[n]]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 24 2009 *)

PARI

apply( A112045=n->(n=prime(n))-sqrtint(n), [1..200]) \\ M. F. Hasler, Oct 19 2018

Python

from math import isqrt
from sympy import prime
def A112045(n): return (p:=prime(n))-isqrt(p) # Chai Wah Wu, Jun 05 2025

Formula

a(n) = A028391(A000040(n)), where A028391(x) = x - floor(sqrt(x)).

a(n) ~ 6/Pi^2 * n log n. - Charles R Greathouse IV, May 29 2013

Extensions

Erroneous name corrected by Antti Karttunen, Jun 03 2014