A069279 - OEIS

A069279

Products of exactly 18 primes (generalization of semiprimes).

%I #37 Aug 24 2024 10:51:53

%S 262144,393216,589824,655360,884736,917504,983040,1327104,1376256,

%T 1441792,1474560,1638400,1703936,1990656,2064384,2162688,2211840,

%U 2228224,2293760,2457600,2490368,2555904,2985984,3014656,3096576,3211264,3244032,3317760,3342336,3440640

%N Products of exactly 18 primes (generalization of semiprimes).

%C Product of 18 not necessarily distinct primes.

%C Divisible by exactly 18 prime powers (not including 1).

%H D. W. Wilson, <a href="/A069279/b069279.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 = 18.

%t Select[Range[31*10^5],PrimeOmega[#]==18&] (* _Harvey P. Dale_, Apr 05 2015 *)

%o (PARI) k=18; start=2^k; finish=4000000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

%o (Python)

%o from math import prod, isqrt

%o from sympy import primerange, integer_nthroot, primepi

%o def A069279(n):

%o 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)))

%o 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,0,1,1,18)))

%o kmin, kmax = 1,2

%o while f(kmax) >= kmax:

%o kmax <<= 1

%o while True:

%o kmid = kmax+kmin>>1

%o if f(kmid) < kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o if kmax-kmin <= 1:

%o break

%o return kmax # _Chai Wah Wu_, Aug 23 2024

%Y Cf. A101637, A101638, A101605, A101606.

%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), 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), this sequence (r = 18), A069280 (r = 19), A069281 (r = 20). - _Jason Kimberley_, Oct 02 2011

%K nonn

%O 1,1

%A _Rick L. Shepherd_, Mar 13 2002