cp's OEIS Frontend

A046316 Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

%I A046316 #35 Oct 18 2024 13:26:31
%S A046316 27,45,63,75,99,105,117,125,147,153,165,171,175,195,207,231,245,255,
%T A046316 261,273,275,279,285,325,333,343,345,357,363,369,385,387,399,423,425,
%U A046316 429,435,455,465,475,477,483,507,531,539,549,555,561,575,595,603,605
%N A046316 Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.
%H A046316 Reinhard Zumkeller, <a href="/A046316/b046316.txt">Table of n, a(n) for n = 1..10000</a>
%o A046316 (Haskell)
%o A046316 a046316 n = a046316_list !! (n-1)
%o A046316 a046316_list = filter ((== 3) . a001222) [1, 3 ..]
%o A046316 -- _Reinhard Zumkeller_, May 05 2015
%o A046316 (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 A046316 (Python)
%o A046316 from math import isqrt
%o A046316 from sympy import primepi, primerange, integer_nthroot
%o A046316 def A046316(n):
%o A046316     def bisection(f,kmin=0,kmax=1):
%o A046316         while f(kmax) > kmax: kmax <<= 1
%o A046316         while kmax-kmin > 1:
%o A046316             kmid = kmax+kmin>>1
%o A046316             if f(kmid) <= kmid:
%o A046316                 kmax = kmid
%o A046316             else:
%o A046316                 kmin = kmid
%o A046316         return kmax
%o A046316     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 A046316     return bisection(f,n,n) # _Chai Wah Wu_, Oct 18 2024
%Y A046316 A369979 sorted into ascending order.
%Y A046316 Subsequence of A014612 and of A046340.
%Y A046316 Cf. A255646 (final digits), A369054, A369058 (characteristic function), A369252 [= A003415(a(n))].
%Y A046316 Subsequences: A046389, A046373, A046405, A075814, A338469, A338556, A338557, A369246.
%K A046316 nonn,easy
%O A046316 1,1
%A A046316 _Patrick De Geest_, Jun 15 1998
%E A046316 Definition clarified by _N. J. A. Sloane_, Dec 19 2017