A116430 The number of n-almost primes less than or equal to 10^n, starting with a(0)=1.

1, 4, 34, 247, 1712, 11185, 68963, 409849, 2367507, 13377156, 74342563, 407818620, 2214357712, 11926066887, 63809981451, 339576381990, 1799025041767, 9494920297227, 49950199374227, 262036734664892

Offset

0,2

Comments

If instead we asked for those less than or equal to 2^n, then the sequence is A000012.

Links

Table of n, a(n) for n=0..19.

Mathematica

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[ AlmostPrimePi[n, 10^n], {n, 0, 13}]

Prog

(PARI)
almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(10^n, n)); \\ Daniel Suteu, Jul 10 2023
(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A116430(n):
    if n<=1: return 3*n+1
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b, isqrt(x//c)+1), a)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b, integer_nthroot(x//c, m)[0]+1), a) for d in g(x, a2, b2, c*b2, m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n, 0, 1, 1, n))) # Chai Wah Wu, Aug 23 2024

Crossrefs

Cf. A036352, A114106, A114453.

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

Cf. A006880, A036352, A109251, A114106, A114453, A120047 - A120053.

Sequence in context: A085695 A049293 A198687 * A216239 A335667 A201684

Adjacent sequences: A116427 A116428 A116429 * A116431 A116432 A116433

Keyword

nonn

Author

Robert G. Wilson v, Feb 10 2006, Jun 01 2006

Extensions

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar

a(15)-a(16) from Donovan Johnson, Oct 01 2010

a(17)-a(19) from Daniel Suteu, Jul 10 2023

Status

approved