A078845 Where 7^n occurs in n-almost-primes, starting at a(0)=1.

1, 4, 17, 82, 385, 1688, 7089, 28893, 115180, 450906, 1740244, 6640747, 25115604, 94312569, 352110321, 1308256678, 4841115048, 17852264639, 65636109307, 240689877440, 880582139867

Offset: 0

Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

• nonn

Comments

A k-almost-prime is a positive integer that has exactly k prime factors, counted with multiplicity.

Examples

			a(2) = 17 since 7^2 is the 17th 2-almost-prime: {4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,...}.
		

Links

• Eric Weisstein's World of Mathematics, Almost Prime.

Crossrefs

Cf. A078840, A078841, A078842, A078843, A078844, A078846.

Programs

Mathematica

l = Table[0, {30}]; e = 0; Do[f = Plus @@ Last /@ FactorInteger[n]; l[[f+1]]++; If[n == 7^e, Print[l[[f+1]]]; e++ ], {n, 1, 7^10}] (* Ryan Propper, Aug 08 2005 *)
AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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, 7^n], {n, 2, 15}] (* Robert G. Wilson v, Feb 09 2006 *)

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def almostprimepi(n,k):
    if k==0: return int(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(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))
def A078845(n): return almostprimepi(7**n,n) if n else 1 # Chai Wah Wu, Oct 02 2024

Extensions

a(7)-a(10) from Ryan Propper, Aug 08 2005

a(11)-a(15) from Robert G. Wilson v, Feb 09 2006

a(16)-a(20) from Donovan Johnson, Sep 27 2010