A116433 -id:A116433 - cp's OEIS frontend

A116434 Consider the array T(r,c), the number of c-almost primes less than or equal to r^c. This is the diagonal T(r,r-1).

0, 1, 3, 13, 90, 726, 7089, 78369, 973404, 13377156, 201443165, 3297443264, 58304208767, 1107693755122

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Feb 15 2006

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 + 1)^n], {n, 11}]

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116434(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+1)**n,n) # Chai Wah Wu, Sep 02 2024

Name rephrased by R. J. Mathar, Jun 20 2021

a(13)-a(14) from Max Alekseyev, Oct 12 2023

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

Original entry on oeis.org

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

Offset: 1

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

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).

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

Cf. A116433, A116434.

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}]

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

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

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

cp's OEIS Frontend

A116434 Consider the array T(r,c), the number of c-almost primes less than or equal to r^c. This is the diagonal T(r,r-1).

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