OFFSET
1,1
COMMENTS
Product of 20 not necessarily distinct primes.
Divisible by exactly 20 prime powers (not including 1).
Any 20-almost prime can be represented in several ways as a product of two 10-almost primes A046314; in several ways as a product of four 5-almost primes A014614; in several ways as a product of five 4-almost primes A014613; and in several ways as a product of ten semiprimes A001358. - Jonathan Vos Post, Dec 12 2004
LINKS
D. W. Wilson, 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 = 20.
a(n) = A078840(20,n). - R. J. Mathar, Jan 30 2019
MATHEMATICA
Select[Range[2*9!, 5*10! ], Plus@@Last/@FactorInteger[ # ]==20 &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
PROG
(PARI) k=20; start=2^k; finish=15000000; v=[]; for(n=start, finish, if(bigomega(n)==k, v=concat(v, n))); v \\ Depending upon the size of k and how many terms are needed, a much more efficient algorithm than the brute-force method above may be desirable. See additional comments in this section of A069280.
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A069281(n):
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)))
def f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, 20)))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
return kmax # Chai Wah Wu, Aug 23 2024
CROSSREFS
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), A014614 (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), this sequence (r = 20). - Jason Kimberley, Oct 02 2011
KEYWORD
nonn
AUTHOR
Rick L. Shepherd, Mar 13 2002
STATUS
approved