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A084127 Prime factor >= other prime factor of n-th semiprime.

%I A084127 #36 Feb 16 2025 08:32:49
%S A084127 2,3,3,5,7,5,7,11,5,13,11,17,7,19,13,23,7,17,11,19,29,31,13,23,37,11,
%T A084127 41,17,43,29,13,31,47,19,53,37,23,59,17,11,61,41,43,19,67,47,71,13,29,
%U A084127 73,31,79,53,23,83,13,59,89,61,37,17,97,67,101,29,41,103,19,71,107,43,31
%N A084127 Prime factor >= other prime factor of n-th semiprime.
%C A084127 Largest nontrivial divisor of n-th semiprime. [_Juri-Stepan Gerasimov_, Apr 18 2010]
%C A084127 Greater of the prime factors of A001358(n). - _Jianing Song_, Aug 05 2022
%H A084127 Zak Seidov, <a href="/A084127/b084127.txt">Table of n, a(n) for n = 1..1000</a>
%H A084127 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>
%F A084127 a(n) = A006530(A001358(n)).
%F A084127 a(n) = A001358(n)/A020639(A001358(n)). [corrected by _Michel Marcus_, Jul 18 2020]
%F A084127 a(n) = A001358(n)/A084126(n).
%t A084127 FactorInteger[#][[-1, 1]]& /@ Select[Range[1000], PrimeOmega[#] == 2&] (* _Jean-François Alcover_, Nov 17 2021 *)
%o A084127 (Haskell)
%o A084127 a084127 = a006530 . a001358  -- _Reinhard Zumkeller_, Nov 25 2012
%o A084127 (PARI) lista(nn) = {for (n=2, nn, if (bigomega(n)==2, f = factor(n); print1(f[length(f~),1], ", ")););} \\ _Michel Marcus_, Jun 05 2013
%o A084127 (Python)
%o A084127 from math import isqrt
%o A084127 from sympy import primepi, primerange, primefactors
%o A084127 def A084127(n):
%o A084127     def bisection(f,kmin=0,kmax=1):
%o A084127         while f(kmax) > kmax: kmax <<= 1
%o A084127         while kmax-kmin > 1:
%o A084127             kmid = kmax+kmin>>1
%o A084127             if f(kmid) <= kmid:
%o A084127                 kmax = kmid
%o A084127             else:
%o A084127                 kmin = kmid
%o A084127         return kmax
%o A084127     def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
%o A084127     return max(primefactors(bisection(f,n,n))) # _Chai Wah Wu_, Oct 23 2024
%Y A084127 Cf. A001358 (the semiprimes), A084126 (lesser of the prime factors of the semiprimes).
%Y A084127 Cf. A014673, A061299, A068318, A087718, A087794, A089994, A089995, A096932, A106550, A106554, A108542, A126663, A131284, A138510, A138511.
%K A084127 nonn
%O A084127 1,1
%A A084127 _Reinhard Zumkeller_, May 15 2003
%E A084127 Corrected by _T. D. Noe_, Nov 15 2006