A116435 Number of n-almost primes less than or equal to n^n.

0, 1, 5, 34, 269, 2613, 28893, 359110, 4934952, 74342563, 1217389949, 21533211312, 409230368646, 8318041706593

Offset

1,3

Comments

Consider the array T(r,c) where is the number of c-almost primes less than or equal to r^c. This is the diagonal T(r,r).

Links

Table of n, a(n) for n=1..14.

Example

a(3)=5 because there are five 3-almost primes <= 27, 8,12,18,20&27.

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 *)
Do[ Print@ AlmostPrimePi[n, n^n], {n, 13}]

Prog

(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116435(n):
    def almostprimepi(n, k):
        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(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
    return almostprimepi(n**n, n) # Chai Wah Wu, Sep 01 2024

Crossrefs

Cf. A116433, A116434.

Sequence in context: A365183 A371391 A058248 * A292877 A257887 A090367

Adjacent sequences: A116432 A116433 A116434 * A116436 A116437 A116438

Keyword

hard,more,nonn

Author

Jonathan Vos Post and Robert G. Wilson v, Feb 15 2006

Extensions

a(13)-a(14) from Donovan Johnson, Oct 05 2010

Definition of T(r,c) corrected by R. J. Mathar, Jun 20 2021

Status

approved