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Numbers with exactly three prime factors, not all distinct.
10

%I #51 Oct 20 2024 13:43:51

%S 8,12,18,20,27,28,44,45,50,52,63,68,75,76,92,98,99,116,117,124,125,

%T 147,148,153,164,171,172,175,188,207,212,236,242,244,245,261,268,275,

%U 279,284,292,316,325,332,333,338,343,356,363,369,387,388,404,412,423,425,428,436,452

%N Numbers with exactly three prime factors, not all distinct.

%C Cubes of primes together with products of a prime and the square of a different prime.

%C Numbers k for which A001222(k) = 3, but A001221(k) < 3. - _Antti Karttunen_, Apr 20 2017

%H Antti Karttunen, <a href="/A285508/b285508.txt">Table of n, a(n) for n = 1..10000</a>

%H Kalle Siukola, <a href="/A285508/a285508.txt">Python program</a>

%p N:= 1000: # for terms <= N

%p P:= select(isprime, [2,seq(i,i=3..N/4,2)]): nP:= nops(P):

%p sort(select(`<=`,[seq(seq(P[i]*P[j]^2,i=1..nP),j=1..nP)],N)); # _Robert Israel_, Oct 20 2024

%t Select[Range[452], PrimeOmega[#] == 3 && PrimeNu[#] < 3 &] (* _Giovanni Resta_, Apr 20 2017 *)

%o (PARI)

%o isA285508(n) = ((omega(n) < 3) && (bigomega(n) == 3));

%o n=0; k=1; while(k <= 10000, n=n+1; if(isA285508(n),write("b285508.txt", k, " ", n);k=k+1));

%o \\ _Antti Karttunen_, Apr 20 2017

%o (Scheme, with my IntSeq-library) (define A285508 (MATCHING-POS 1 1 (lambda (n) (and (= 3 (A001222 n)) (< (A001221 n) 3))))) ;; _Antti Karttunen_, Apr 20 2017

%o (Python)

%o from sympy import primefactors, primeomega

%o def omega(n): return len(primefactors(n))

%o def bigomega(n): return primeomega(n)

%o print([n for n in range(1, 501) if omega(n)<3 and bigomega(n) == 3]) # _Indranil Ghosh_, Apr 20 2017 and _Kalle Siukola_, Oct 25 2023

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot

%o def A285508(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 f(x): return int(n+x-sum(primepi(x//(k**2))-(a<<1)+primepi(isqrt(x//k))-1 for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1))))

%o return bisection(f,n,n) # _Chai Wah Wu_, Oct 20 2024

%Y Cf. A001221, A001222.

%Y Setwise difference of A014612 and A007304.

%Y Union of A030078 and A054753.

%K easy,nonn

%O 1,1

%A _Kalle Siukola_, Apr 20 2017