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

1,1

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

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

Mathematica

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

Prog

(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

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

Sequence in context: A090052 A163285 A036329 * A046371 A348824 A175162

Adjacent sequences: A014611 A014612 A014613 * A014615 A014616 A014617

Keyword

nonn,changed

Author

Eric W. Weisstein

Extensions

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

Status

approved