cp's OEIS Frontend

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

%I A014614 #34 Feb 16 2025 08:32:33
%S A014614 32,48,72,80,108,112,120,162,168,176,180,200,208,243,252,264,270,272,
%T A014614 280,300,304,312,368,378,392,396,405,408,420,440,450,456,464,468,496,
%U A014614 500,520,552,567,588,592,594,612,616,630,656,660,675,680,684,688,696
%N A014614 Numbers that are products of 5 primes (or 5-almost primes, a generalization of semiprimes).
%C A014614 Divisible by exactly 5 prime powers (not including 1).
%H A014614 T. D. Noe, <a href="/A014614/b014614.txt">Table of n, a(n) for n = 1..10000</a>
%H A014614 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>
%F A014614 Product p_i^e_i with sum e_i = 5.
%F A014614 a(n) ~ 24n log n/(log log n)^4. - _Charles R Greathouse IV_, Mar 20 2013
%F A014614 a(n) = A078840(5,n). - _R. J. Mathar_, Jan 30 2019
%t A014614 Select[Range[300], Plus @@ Last /@ FactorInteger[ # ] == 5 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%o A014614 (PARI) is(n)=bigomega(n)==5 \\ _Charles R Greathouse IV_, Mar 20 2013
%o A014614 (Python)
%o A014614 from math import isqrt
%o A014614 from sympy import primepi, primerange, integer_nthroot
%o A014614 def A014614(n):
%o A014614     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 A014614     m, k = n, f(n)
%o A014614     while m != k:
%o A014614         m, k = k, f(k)
%o A014614     return m # _Chai Wah Wu_, Aug 17 2024
%Y A014614 Cf. A046304, A114453 (number of 5-almost primes <= 10^n).
%Y A014614 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 A014614 nonn
%O A014614 1,1
%A A014614 _Eric W. Weisstein_
%E A014614 More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu) and _Patrick De Geest_, Jun 15 1998