A046314 - OEIS

A046314

Numbers that are divisible by exactly 10 primes with multiplicity.

%I #41 Nov 04 2024 09:31:03

%S 1024,1536,2304,2560,3456,3584,3840,5184,5376,5632,5760,6400,6656,

%T 7776,8064,8448,8640,8704,8960,9600,9728,9984,11664,11776,12096,12544,

%U 12672,12960,13056,13440,14080,14400,14592,14848,14976,15872,16000,16640

%N Numbers that are divisible by exactly 10 primes with multiplicity.

%C Also called 10-almost primes. Products of exactly 10 primes (not necessarily distinct). Any 10-almost prime can be represented in several ways as a product of two 5-almost primes A014614 and in several ways as a product of five semiprimes A001358. - _Jonathan Vos Post_, Dec 11 2004

%H T. D. Noe, <a href="/A046314/b046314.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">Reference</a>

%F Product p_i^e_i with Sum e_i = 10.

%F a(n) ~ 362880n log n / (log log n)^9. - _Charles R Greathouse IV_, May 06 2013

%t Select[Range[5000], Plus @@ Last /@ FactorInteger[ # ] == 10 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)

%t Select[Range[17000],PrimeOmega[#]==10&] (* _Harvey P. Dale_, Jun 23 2018 *)

%o (PARI) is(n)=bigomega(n)==10 \\ _Charles R Greathouse IV_, Mar 21 2013

%o (Python)

%o from math import isqrt, prod

%o from sympy import primerange, integer_nthroot, primepi

%o def A046314(n):

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%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+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,10)))

%o return bisection(f,n,n) # _Chai Wah Wu_, Nov 03 2024

%Y Cf. A046313, A120051 (number of 10-almost primes <= 10^n).

%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), this sequence (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 _Patrick De Geest_, Jun 15 1998