A072000 -id:A072000 - cp's OEIS frontend

A247558 Smallest integer x > 0 such that the number of semiprimes in the interval (x/2, x] equals n.

4, 6, 10, 15, 25, 26, 35, 38, 39, 57, 58, 62, 65, 86, 87, 91, 94, 95, 121, 122, 123, 134, 142, 143, 145, 146, 159, 161, 169, 202, 203, 205, 206, 209, 214, 215, 217, 218, 219, 221, 262, 265, 278, 299, 301, 302, 303, 305, 309, 326, 327, 329, 335, 341, 346, 361, 362, 365, 382, 386, 393, 394, 395, 398

Offset: 1

Jonathan Vos Post and Robert G. Wilson v, Sep 19 2014

nonn

Analogous to A080359: the Labos Elemer primes.

			a(6) = 26 because in the interval, (13, 26], {14, 15, 21, 22, 25, 26} are six semiprimes.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A001358, A072000, A080359.

SemiPrimeQ[n_] := PrimeOmega[n] == 2; mx = 1000; t = Table[0, {mx + 1}]; s = 0; Do[ If[ SemiPrimeQ[k], s++]; If[ SemiPrimeQ[k/2], s--]; If[s <= mx && t[[s + 1]] == 0, t[[s + 1]] = k], {k, 8*mx}]; Rest[t]

A334940 Partial sums of A230595.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 3, 4, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 12, 14, 14, 14, 15, 17, 17, 17, 17, 17, 17, 17, 19, 21, 23, 23, 23, 25, 27, 27, 27, 27, 27, 27, 27, 29, 29, 29, 30, 30, 32, 32, 32, 32, 34, 34, 36, 38, 38, 38, 38, 40, 40, 40, 42, 42, 42, 42, 44, 44, 44, 44, 44, 46

Offset: 1

Daniel Suteu, May 17 2020

nonn

Sum of the Dirichlet convolution of the characteristic function of primes (A010051) with itself from 1 to n.

(a(n) + A000720(floor(sqrt(n))))/2 equals the number of semiprimes <= n.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000720, A001222, A010051, A072000, A230595.

a:= proc(n) option remember; `if`(n<4, 0, a(n-1) +
     `if`(numtheory[bigomega](n)=2, `if`(issqr(n), 1, 2), 0))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 20 2020

f[n_] := DivisorSum[n, 1 &, PrimeQ[#] && PrimeQ[n/#] &]; Accumulate @ Array[f, 100] (* Amiram Eldar, May 20 2020 *)

a(n) = my(s=sqrtint(n)); 2*sum(k=1, s, if(isprime(k), primepi(n\k), 0)) - primepi(s)^2;

from math import isqrt
from sympy import primepi, prime
def A334940(n): return (int(sum(primepi(n//prime(k))-k+1 for k in range(1,primepi(isqrt(n))+1)))<<1) - primepi(isqrt(n)) # Chai Wah Wu, Jul 23 2024

a(n) = Sum_{k=1..n} Sum_{d|k} A010051(d) * A010051(k/d).
a(n) = 2*Sum_{p prime <= sqrt(n)} A000720(floor(n/p)) - A000720(floor(sqrt(n)))^2.
a(n) = 2*A072000(n) - A000720(floor(sqrt(n))).
a(n) = 2*A072613(n) + A000720(floor(sqrt(n))). - Vaclav Kotesovec, May 21 2020
a(n) ~ 2*n*log(log(n))/log(n). - Vaclav Kotesovec, May 21 2020

A108216 Number of semiprimes between 10n and 10n + 9.

Original entry on oeis.org

3, 3, 4, 5, 2, 4, 3, 2, 4, 4, 1, 4, 4, 2, 5, 3, 3, 2, 3, 1, 6, 6, 2, 2, 2, 3, 3, 2, 2, 4, 5, 2, 5, 3, 2, 2, 3, 2, 3, 5, 2, 4, 2, 1, 3, 4, 2, 3, 4, 2, 3, 5, 3, 4, 3, 4, 3, 2, 4, 2, 0, 2, 4, 3, 1, 1, 3, 3, 4, 5, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 4, 0, 3, 4, 1, 4, 3, 2, 4, 2, 3, 4, 3, 2, 4, 2, 3, 3, 3, 3, 2, 1, 1, 5

Offset: 0

Giovanni Teofilatto, Jun 16 2005

nonn

a(60) = a(82) = a(142) = 0. - Jonathan Vos Post, Jun 16 2005

			a(0) = 3 because between 0 and 9 there are three semiprimes: 4, 6 and 9.
a(1) = 3 because between 10 and 19 there are three semiprimes: 10, 14 and 15.

Cf. A038800 number of primes between 10n and 10n+9.

Cf. A064911, A066265, A072000.

f[n_] := Sum[ PrimePi[n/Prime[i]] - (i - 1), {i, PrimePi[ Sqrt[n]]}]; Table[f[10n + 9] - f[Max[10n - 1, 0]], {n, 0, 104}] (* Robert G. Wilson v, Ray Chandler *)

Edited and extended by Ray Chandler, Jul 07 2005

A115854 Difference between number of semiprimes <= 10^n and the asymptotic approximation round(10^n*loglog(10^n)/log(10^n)).

Original entry on oeis.org

0, 0, 0, 19, 214, 2154, 19974, 179590, 1610937, 14515403, 131560754, 1199914216, 11009605949, 101581094033, 942018562525

Offset: 0

Jonathan Vos Post, Mar 14 2006

nonn

			a(5) = A036352(5) - round(...10^5...) = 23378 - 21224 = 2154.

Cf. A001358, A036352, A066265, A072000.

Edited by Don Reble, Mar 29 2006

cp's OEIS Frontend

A247558 Smallest integer x > 0 such that the number of semiprimes in the interval (x/2, x] equals n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A334940 Partial sums of A230595.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A108216 Number of semiprimes between 10n and 10n + 9.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A115854 Difference between number of semiprimes <= 10^n and the asymptotic approximation round(10^n*loglog(10^n)/log(10^n)).

Views

Author

Keywords

Examples

Crossrefs

Extensions