A234102 Integers of the form (p*q*r + 1)/2, where p, q, r are distinct primes.
53, 83, 98, 116, 128, 137, 143, 173, 179, 193, 200, 215, 218, 228, 233, 242, 278, 281, 298, 305, 308, 314, 323, 326, 332, 333, 353, 358, 371, 380, 389, 398, 403, 431, 443, 449, 452, 458, 468, 479, 485, 494, 501, 503, 508, 512, 523, 533, 543, 548, 553, 557
Offset: 1
Examples
53 = (3*5*7 + 1)/2.
Programs
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Mathematica
t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t + 1)/2, 120] (* A234102 *) v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}] (* A234103 *) (w + 1)/2 (* A234104 *) (* Peter J. C. Moses, Dec 23 2013 *) -
Python
from math import isqrt from sympy import primepi, primerange, integer_nthroot def A234102(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax 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))) return bisection(f,n,n)+1>>1 # Chai Wah Wu, Oct 18 2024
Formula
1 + A234099.
a(n) = (A046389(n)+1)/2. - Chai Wah Wu, Oct 18 2024