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A128301
Indices of squares (of primes) in the semiprimes.
20
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
1,2
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
EXAMPLE
a(4) = 17 as 49 = 7^2 = prime(4)^2, the fourth square in the semiprimes, is the seventeenth semiprime.
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Rick L. Shepherd, Feb 25 2007
STATUS
approved