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A347044 Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

%I A347044 #26 Apr 13 2025 14:54:47
%S A347044 1,2,3,2,5,3,7,4,3,5,11,6,13,7,5,4,17,9,19,10,7,11,23,6,5,13,9,14,29,
%T A347044 15,31,8,11,17,7,9,37,19,13,10,41,21,43,22,15,23,47,12,7,25,17,26,53,
%U A347044 9,11,14,19,29,59,15,61,31,21,8,13,33,67,34,23,35,71
%N A347044 Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.
%C A347044 Appears to contain each positive integer at least once, but only a finite number of times.
%H A347044 Amiram Eldar, <a href="/A347044/b347044.txt">Table of n, a(n) for n = 1..10000</a>
%F A347044 a(n) = Product_{k=floor(A001222(n)/2)+1..A001222(n)} A027746(n,k). - _Amiram Eldar_, Nov 02 2024
%e A347044 The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 7716.
%t A347044 Table[Max[Select[Divisors[n],PrimeOmega[#]==Ceiling[PrimeOmega[n]/2]&]],{n,100}]
%t A347044 a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; Times @@ p[[Floor[np/2] + 1;; np]]]; Array[a, 100] (* _Amiram Eldar_, Nov 02 2024 *)
%o A347044 (Python)
%o A347044 from sympy import divisors, factorint
%o A347044 def a(n):
%o A347044     npf = len(factorint(n, multiple=True))
%o A347044     for d in divisors(n)[::-1]:
%o A347044         if len(factorint(d, multiple=True)) == (npf+1)//2: return d
%o A347044     return 1
%o A347044 print([a(n) for n in range(1, 72)]) # _Michael S. Branicky_, Aug 18 2021
%o A347044 (Python)
%o A347044 from math import prod
%o A347044 from sympy import factorint
%o A347044 def A347044(n):
%o A347044     fs = factorint(n,multiple=True)
%o A347044     l = len(fs)
%o A347044     return prod(fs[l//2:]) # _Chai Wah Wu_, Aug 20 2021
%Y A347044 The greatest divisor without the condition is A006530 (smallest: A020639).
%Y A347044 Divisors of this type are counted by A096825 (exact: A345957).
%Y A347044 The case of powers of 2 is A163403.
%Y A347044 The smallest divisor of this type is given by A347043 (exact: A347045).
%Y A347044 The exact version is A347046.
%Y A347044 A000005 counts divisors.
%Y A347044 A001221 counts distinct prime factors.
%Y A347044 A001222 counts all prime factors (also called bigomega).
%Y A347044 A038548 counts inferior (or superior) divisors (strict: A056924).
%Y A347044 A056239 adds up prime indices, row sums of A112798.
%Y A347044 A207375 lists central divisors (min: A033676, max: A033677).
%Y A347044 A340387 lists numbers whose sum of prime indices is twice bigomega.
%Y A347044 A340609 lists numbers whose maximum prime index divides bigomega.
%Y A347044 A340610 lists numbers whose maximum prime index is divisible by bigomega.
%Y A347044 A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
%Y A347044 Cf. A000720, A027746, A060775, A106529, A217581, A244990, A335433, A335448.
%K A347044 nonn
%O A347044 1,2
%A A347044 _Gus Wiseman_, Aug 16 2021