A078844 Where 5^n occurs in n-almost-primes, starting at a(0)=1.

1, 3, 9, 30, 90, 269, 788, 2249, 6340, 17526, 47911, 129639, 348251, 929714, 2469499, 6532869, 17219031, 45246630, 118572805, 309998131, 808746993, 2105893899, 5474080107, 14207001052, 36818679828, 95292132897, 246327403310

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) = 9 since 5^2 is the 9th 2-almost-prime: {4,6,9,10,14,15,21,22,25,...}.
		

Links

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

Crossrefs

Cf. A078840, A078841, A078842, A078843, A078845, A078846.

Programs

Mathematica

l = Table[0, {30}]; e = 0; Do[f = Plus @@ Last /@ FactorInteger[n]; l[[f+1]]++; If[n == 5^e, Print[l[[f+1]]]; e++ ], {n, 1, 5^10}] (* Ryan Propper, Aug 08 2005 *)
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 *)
Join[{1},Table[ AlmostPrimePi[n, 5^n], {n, 1, 25}]] (* Robert G. Wilson v, Feb 10 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 A078844(n): return almostprimepi(5**n, n) if n else 1 # Chai Wah Wu, Nov 07 2024

Extensions

a(8)-a(10) from Ryan Propper, Aug 08 2005

a(11)-a(25) from Robert G. Wilson v, Feb 10 2006

a(26) from Donovan Johnson, Sep 27 2010