A014614 - OEIS

%I #33 Aug 17 2024 22:28:54

%S 32,48,72,80,108,112,120,162,168,176,180,200,208,243,252,264,270,272,

%T 280,300,304,312,368,378,392,396,405,408,420,440,450,456,464,468,496,

%U 500,520,552,567,588,592,594,612,616,630,656,660,675,680,684,688,696

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

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

%H T. D. Noe, <a href="/A014614/b014614.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>

%F Product p_i^e_i with sum e_i = 5.

%F a(n) ~ 24n log n/(log log n)^4. - _Charles R Greathouse IV_, Mar 20 2013

%F a(n) = A078840(5,n). - _R. J. Mathar_, Jan 30 2019

%t Select[Range[300], Plus @@ Last /@ FactorInteger[ # ] == 5 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)

%o (PARI) is(n)=bigomega(n)==5 \\ _Charles R Greathouse IV_, Mar 20 2013

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot

%o def A014614(n):

%o     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)))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Aug 17 2024

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

%Y 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

%K nonn

%O 1,1

%A _Eric W. Weisstein_

%E More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu) and _Patrick De Geest_, Jun 15 1998