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A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

%I A347046 #22 Apr 13 2025 15:01:38
%S A347046 1,1,1,2,1,3,1,1,3,5,1,1,1,7,5,4,1,1,1,1,7,11,1,6,5,13,1,1,1,1,1,1,11,
%T A347046 17,7,9,1,19,13,10,1,1,1,1,1,23,1,1,7,1,17,1,1,9,11,14,19,29,1,15,1,
%U A347046 31,1,8,13,1,1,1,23,1,1,1,1,37,1,1,11,1,1,1,9
%N A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.
%C A347046 Problem: What are the positions of last appearances > 1?
%H A347046 Amiram Eldar, <a href="/A347046/b347046.txt">Table of n, a(n) for n = 1..10000</a>
%F A347046 a(n) = Product_{k=A001222(n)/2+1..A001222(n)} A027746(n,k) if A001222(n) is even, and 1 otherwise. - _Amiram Eldar_, Nov 02 2024
%e A347046 The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 15.
%t A347046 Table[If[#=={},1,Max[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]
%t A347046 a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[np/2+1 ;; np]]]]; Array[a, 100] (* _Amiram Eldar_, Nov 02 2024 *)
%o A347046 (Python)
%o A347046 from sympy import divisors, factorint
%o A347046 def a(n):
%o A347046     npf = len(factorint(n, multiple=True))
%o A347046     for d in divisors(n)[-1:0:-1]:
%o A347046         if 2*len(factorint(d, multiple=True)) == npf: return d
%o A347046     return 1
%o A347046 print([a(n) for n in range(1, 82)]) # _Michael S. Branicky_, Aug 18 2021
%o A347046 (Python)
%o A347046 from math import prod
%o A347046 from sympy import factorint
%o A347046 def A347046(n):
%o A347046     fs = factorint(n,multiple=True)
%o A347046     q, r = divmod(len(fs),2)
%o A347046     return 1 if r else prod(fs[q:]) # _Chai Wah Wu_, Aug 20 2021
%Y A347046 The greatest divisor without the condition is A006530 (smallest: A020639).
%Y A347046 Positions of 1's are A026424.
%Y A347046 The case of powers of 2 is A072345.
%Y A347046 Positions of first appearances are A123667 (sorted: A123666).
%Y A347046 Divisors of this type are counted by A345957 (rounded: A096825).
%Y A347046 The rounded version is A347044.
%Y A347046 The smallest divisor of this is A347045 (rounded: A347043).
%Y A347046 A000005 counts divisors.
%Y A347046 A001221 counts distinct prime factors.
%Y A347046 A001222 counts all prime factors (also called bigomega).
%Y A347046 A056239 adds up prime indices, row sums of A112798.
%Y A347046 A207375 lists central divisors (min: A033676, max: A033677).
%Y A347046 A340387 lists numbers whose sum of prime indices is twice bigomega.
%Y A347046 A340609 lists numbers whose maximum prime index divides bigomega.
%Y A347046 A340610 lists numbers whose maximum prime index is divisible by bigomega.
%Y A347046 A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
%Y A347046 Cf. A000720, A027746, A038548, A060775, A106529, A244991, A335433, A335448.
%K A347046 nonn
%O A347046 1,4
%A A347046 _Gus Wiseman_, Aug 16 2021