A114453 Number of 5-almost primes less than or equal to 10^n.

0, 0, 4, 76, 963, 11185, 124465, 1349779, 14371023, 150982388, 1570678136, 16218372618, 166497674684, 1701439985694, 17323079621014, 175846040834673, 1780617141307093, 17993699600756449, 181520864946969233

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Example

There are 4 five-almost primes up to 100: 32,48,72 and 80, so a(2) = 4.

Mathematica

FiveAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k*Prime@l)] - l + 1, {i, PrimePi[n^(1/5)]}, {j, i, PrimePi[(n/Prime@i)^(1/4)]}, {k, j, PrimePi[(n/(Prime@i*Prime@j))^(1/3)]}, {l, k, PrimePi[(n/(Prime@i*Prime@j*Prime@k))^(1/2)]}]; Table[ FiveAlmostPrimePi[10^n], {n, 0, 12}]

Prog

(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A114453(n):
    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, 5))) # Chai Wah Wu, Sep 18 2024

Crossrefs

Cf. A006880, A066265, A109251, A014614, A116382.

Sequence in context: A045996 A190395 A240281 * A093184 A214814 A144210

Adjacent sequences: A114450 A114451 A114452 * A114454 A114455 A114456

Keyword

nonn,more

Author

Robert G. Wilson v, Feb 07 2006

Extensions

a(13)-a(14) from Robert G. Wilson v, Jan 07 2007

a(15)-a(18) from Henri Lifchitz, Feb 03 2025

Status

approved