A128301 Indices of squares (of primes) in the semiprimes.

1, 3, 9, 17, 40, 56, 90, 114, 164, 253, 289, 404, 484, 533, 634, 783, 973, 1031, 1233, 1373, 1452, 1683, 1842, 2112, 2483, 2676, 2779, 2995, 3108, 3320, 4124, 4384, 4775, 4926, 5593, 5741, 6172, 6644, 6962, 7448, 7955, 8108, 8978, 9147, 9512, 9697, 10842

Offset: 1

Author

Rick L. Shepherd, Feb 25 2007

Keywords

• nonn

Comments

A001358(a(n)) = A001248(n) = A000040(n)^2.

Numbers n with property that tau(semiprime(n)) is not semiprime. - Juri-Stepan Gerasimov, Oct 15 2010

Examples

			a(4) = 17 as 49 = 7^2 = prime(4)^2, the fourth square in the semiprimes, is the seventeenth semiprime.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A128302, A001358, A001248, A072000.

Programs

Mathematica

With[{sp=Select[Range[50000],PrimeOmega[#]==2&]},Flatten[Table[ Position[ sp,Prime[ n]^2],{n,Floor[Sqrt[Length[sp]]]}]]] (* Harvey P. Dale, Nov 17 2014 *)

PARI

a(n)=my(s=0,i=0); n=prime(n)^2; forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2
\\ Charles R Greathouse IV, Apr 21 2011

Perl

-MMath::Pari=factorint,PARI -wle 'my $c = 0; my $s = PARI 1; while (1) { ++$s; my($sp, $si) = @{factorint($s)}; next if @$sp > 2; next if $si->[0] + (@$si > 1 ? $si->[1] : 0) != 2; ++$c; print "$s => $c" if @$sp == 1}' # Hugo van der Sanden, Sep 25 2007

Python

from math import isqrt
from sympy import prime, primepi
def A128301(n):
    m = prime(n)**2
    return int(sum(primepi(m//prime(k))-k+1 for k in range(1,n+1))) # Chai Wah Wu, Jul 23 2024