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A234099 Integers of the form (p*q*r - 1)/2, where p, q, r are distinct primes.

%I A234099 #11 Oct 18 2024 11:43:00
%S A234099 52,82,97,115,127,136,142,172,178,192,199,214,217,227,232,241,277,280,
%T A234099 297,304,307,313,322,325,331,332,352,357,370,379,388,397,402,430,442,
%U A234099 448,451,457,467,478,484,493,500,502,507,511,522,532,542,547,552,556
%N A234099 Integers of the form (p*q*r - 1)/2, where p, q, r are distinct primes.
%F A234099 -1 + A234102.
%F A234099 a(n) = (A046389(n)-1)/2. - _Chai Wah Wu_, Oct 18 2024
%e A234099 52 = (3*5*7 - 1)/2.
%t A234099 t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t - 1)/2, 120] (* A234099 *)
%t A234099 v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234100 *)
%t A234099 (w - 1)/2 (* A234101 *)    (* _Peter J. C. Moses_, Dec 23 2013 *)
%o A234099 (Python)
%o A234099 from math import isqrt
%o A234099 from sympy import primepi, primerange, integer_nthroot
%o A234099 def A234099(n):
%o A234099     def bisection(f,kmin=0,kmax=1):
%o A234099         while f(kmax) > kmax: kmax <<= 1
%o A234099         while kmax-kmin > 1:
%o A234099             kmid = kmax+kmin>>1
%o A234099             if f(kmid) <= kmid:
%o A234099                 kmax = kmid
%o A234099             else:
%o A234099                 kmin = kmid
%o A234099         return kmax
%o A234099     def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
%o A234099     return bisection(f,n,n)>>1 # _Chai Wah Wu_, Oct 18 2024
%Y A234099 Cf. A046389, A234100, A234101, A234093, A234102.
%K A234099 nonn,easy
%O A234099 1,1
%A A234099 _Clark Kimberling_, Dec 27 2013