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Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.
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%I #35 Oct 18 2024 13:26:31

%S 27,45,63,75,99,105,117,125,147,153,165,171,175,195,207,231,245,255,

%T 261,273,275,279,285,325,333,343,345,357,363,369,385,387,399,423,425,

%U 429,435,455,465,475,477,483,507,531,539,549,555,561,575,595,603,605

%N Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

%H Reinhard Zumkeller, <a href="/A046316/b046316.txt">Table of n, a(n) for n = 1..10000</a>

%o (Haskell)

%o a046316 n = a046316_list !! (n-1)

%o a046316_list = filter ((== 3) . a001222) [1, 3 ..]

%o -- _Reinhard Zumkeller_, May 05 2015

%o (PARI) list(lim)=my(v=List(),pq); forprime(p=3,lim\9, forprime(q=3,min(lim\3\p,p), pq=p*q; forprime(r=3,lim\pq, listput(v, pq*r)))); Set(v) \\ _Charles R Greathouse IV_, Aug 23 2017

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot

%o def A046316(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*m))-b+1 for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))

%o return bisection(f,n,n) # _Chai Wah Wu_, Oct 18 2024

%Y A369979 sorted into ascending order.

%Y Subsequence of A014612 and of A046340.

%Y Cf. A255646 (final digits), A369054, A369058 (characteristic function), A369252 [= A003415(a(n))].

%Y Subsequences: A046389, A046373, A046405, A075814, A338469, A338556, A338557, A369246.

%K nonn,easy

%O 1,1

%A _Patrick De Geest_, Jun 15 1998

%E Definition clarified by _N. J. A. Sloane_, Dec 19 2017