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

1,3

Links

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

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

Prog

(Python)
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

Crossrefs

Cf. A116433, A116435.

Sequence in context: A097711 A114477 A345104 * A174290 A034513 A257661

Adjacent sequences: A116431 A116432 A116433 * A116435 A116436 A116437

Keyword

hard,more,nonn

Author

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

Extensions

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

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

Status

approved