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Numbers with at most 3 prime factors (counted with multiplicity).
20

%I #20 Aug 23 2024 21:11:07

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,27,28,

%T 29,30,31,33,34,35,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,

%U 58,59,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,85,86

%N Numbers with at most 3 prime factors (counted with multiplicity).

%C Complement of A033987: A001222(a(n))<=3; A117358(a(n))=1. - _Reinhard Zumkeller_, Mar 10 2006

%C Also numbers such that exist permutations of all proper divisors only with coprime adjacent elements: A178254(a(n))>0. - _Reinhard Zumkeller_, May 24 2010

%H Klaus Brockhaus, <a href="/A037144/b037144.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ 2n log n/(log log n)^2. - _Charles R Greathouse IV_, Sep 14 2015

%t Select[Range[100],PrimeOmega[#]<4&] (* _Harvey P. Dale_, Oct 15 2015 *)

%o (Magma) [ n: n in [1..86] | n eq 1 or &+[ t[2]: t in Factorization(n) ] le 3 ]; /* Klaus Brockhaus, Mar 20 2007 */

%o (PARI) is(n)=bigomega(n)<4 \\ _Charles R Greathouse IV_, Sep 14 2015

%o (Python)

%o from math import prod, isqrt

%o from sympy import primerange, integer_nthroot, primepi

%o def A037144(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+x-2-primepi(x)-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,4)))

%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 A037143 is a subsequence.

%Y Cf. A033987, A001222, A117358, A128644.

%K nonn

%O 1,2

%A _N. J. A. Sloane_.

%E More terms from _Reinhard Zumkeller_, Mar 10 2006

%E More terms from _Klaus Brockhaus_, Mar 20 2007