cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A066265 a(n) = number of semiprimes < 10^n.

Original entry on oeis.org

0, 3, 34, 299, 2625, 23378, 210035, 1904324, 17427258, 160788536, 1493776443, 13959990342, 131126017178, 1237088048653, 11715902308080, 111329817298881, 1061057292827269, 10139482913717352, 97123037685177087, 932300026230174178, 8966605849641219022, 86389956293761485464, 833671466551239927908, 8056846659984852885191
Offset: 0

Views

Author

Patrick De Geest, Dec 10 2001

Keywords

Comments

Apart from the first nonzero term the sequence is identical to A036352. - Hugo Pfoertner, Jul 22 2003

Examples

			Below 10 there are three semiprimes: 4 (2*2), 6 (2*3) and 9 (3*3).

Links

Crossrefs

Cf. A001358, A064911, A072000, A036352 (identical starting from a(2)), A220262, A292785.

Programs

  • Mathematica
    f[n_] := Sum[ PrimePi[(10^n - 1)/Prime[i]], {i, PrimePi[ Sqrt[10^n]]}] - Binomial[ PrimePi[ Sqrt[10^n]], 2]; Do[ Print[ f[n]], {n, 0, 14}] (* Robert G. Wilson v, May 16 2005 *)
SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}]; Array[ SemiPrimePi[10^# - 1] &, 14, 0] (* Robert G. Wilson v, Jan 21 2015 *)
  • PARI
    a(n)=my(s);forprime(p=2,sqrt(10^n),s+=primepi((10^n-1)\p)); s-binomial(primepi(sqrt(10^n)),2) \\ Charles R Greathouse IV, Apr 23 2012
  • Perl
    use Math::Prime::Util qw/:all/; use integer; sub countsp { my($k,$sum,$pc)=($[0]-1,0,1); prime_precalc(60_000_000); forprimes { $sum += prime_count($k/$) + 1 - $pc++; } int(sqrt($k)); $sum; } foreach my $n (0..16) { say "$n: ", countsp(10**$n); } # Dana Jacobsen, May 11 2014
  • Python
    from math import isqrt
from sympy import primepi, primerange
def A066265(n): return int((-(t:=primepi(s:=isqrt(m:=10**n)))*(t-1)>>1)+sum(primepi(m//k) for k in primerange(1, s+1))) if n>1 else 3*n # Chai Wah Wu, Aug 16 2024

Formula

(1/2)*( pi(10^(n/2)) + Sum_{i=1..pi(10^n)} pi( (10^n-1)/P_i) ) = Sum_{i=1..pi(sqrt(10^n))} pi( (10^n-1)/P_i ) - binomial( pi(sqrt(10^n)), 2). - Robert G. Wilson v, May 16 2005

Extensions

More terms from Hugo Pfoertner, Jul 22 2003
a(14) from Robert G. Wilson v, May 16 2005
a(15)-a(16) from Donovan Johnson, Mar 18 2010
a(17)-a(18) from Dana Jacobsen, May 11 2014
a(19)-a(21) from Henri Lifchitz, Jul 04 2015
a(22)-a(23) from Henri Lifchitz, Nov 09 2024

A085770 Number of odd semiprimes < 10^n. Number of terms of A046315 < 10^n.

Original entry on oeis.org

0, 1, 19, 204, 1956, 18245, 168497, 1555811, 14426124, 134432669, 1258822220, 11840335764, 111817881036, 1059796387004, 10076978543513, 96091983644261, 918679875630905, 8803388145953381, 84537081118605467, 813340036541900706, 7838825925851034479, 75669246175972479567
Offset: 0

Views

Author

Hugo Pfoertner, Jul 22 2003

Keywords

Examples

			a(1)=1 because A046315(1)=9=3*3 is the only odd semiprime < 10^1,
a(2)=19 because there are 19 terms of A046315 < 10^2.

Crossrefs

Cf. A046315 (odd numbers divisible by exactly 2 primes), A066265 (number of semiprimes < 10^n), A220262, A292785.

Programs

  • Mathematica
    OddSemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, 2, PrimePi@ Sqrt@ n}]; Table[ OddSemiPrimePi[10^n], {n, 14}] (* Robert G. Wilson v, Feb 02 2006 *)
  • Python
    from math import isqrt
from sympy import primepi, primerange
def A085770(n): return int((-(t:=primepi(s:=isqrt(m:=10**n)))*(t-1)>>1)+sum(primepi(m//k) for k in primerange(3, s+1))) if n>1 else n # Chai Wah Wu, Oct 17 2024

Formula

a(n) = A066265(n) - A220262(n) for n > 1. - Jinyuan Wang, Jul 30 2021

Extensions

a(10)-a(14) from Robert G. Wilson v, Feb 02 2006
a(15)-a(16) from Donovan Johnson, Mar 18 2010
a(0) inserted by and a(17)-a(21) from Jinyuan Wang, Jul 30 2021

A292785 Number of odd squarefree semiprimes < 10^n.

Original entry on oeis.org

0, 16, 194, 1932, 18181, 168330, 1555366, 14424896, 134429269, 1258812629, 11840308472, 111817802539, 1059796159358, 10076977878935, 96091981692305, 918679869869451, 8803388128870716, 84537081067757934, 813340036390023775, 7838825925395981969, 75669246174605279757
Offset: 1

Views

Author

Hugo Pfoertner, Oct 10 2017

Keywords

Examples

			a(2)=16 because there are 16 squarefree odd semiprimes < 10^2: 15=3*5, 21=3*7, 33=3*11, 35=5*7, 39=3*13, 51=3*17, 55=5*11, 57=3*19, 65=5*13, 69=3*23, 77=7*11, 85=5*17, 87=3*29, 91=7*13, 93=3*31, 95=5*19.

Crossrefs

Cf. A046388, A066265, A122121, A220262.

Programs

Formula

a(n) = A066265(n) - A122121(n) - A220262(n) + 1 for n > 1.

Extensions

a(21) from Jinyuan Wang, Jul 30 2021
Showing 1-3 of 3 results.

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