A014614 Numbers that are products of 5 primes (or 5-almost primes, a generalization of semiprimes).

32, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, 200, 208, 243, 252, 264, 270, 272, 280, 300, 304, 312, 368, 378, 392, 396, 405, 408, 420, 440, 450, 456, 464, 468, 496, 500, 520, 552, 567, 588, 592, 594, 612, 616, 630, 656, 660, 675, 680, 684, 688, 696

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

Divisible by exactly 5 prime powers (not including 1).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Almost Prime

Crossrefs

Cf. A046304, A114453 (number of 5-almost primes <= 10^n).

Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), this sequence (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Programs

Mathematica

Select[Range[300], Plus @@ Last /@ FactorInteger[ # ] == 5 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)

PARI

is(n)=bigomega(n)==5 \\ Charles R Greathouse IV, Mar 20 2013

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A014614(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m*r*s))-d for a,k in enumerate(primerange(integer_nthroot(x,5)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,4)[0]+1),a) for c,r in enumerate(primerange(m,integer_nthroot(x//(k*m),3)[0]+1),b) for d,s in enumerate(primerange(r,isqrt(x//(k*m*r))+1),c)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024

Formula

Product p_i^e_i with sum e_i = 5.

a(n) ~ 24n log n/(log log n)^4. - Charles R Greathouse IV, Mar 20 2013

a(n) = A078840(5,n). - R. J. Mathar, Jan 30 2019

Extensions

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu) and Patrick De Geest, Jun 15 1998