A347044 - OEIS

%I #23 Nov 02 2024 09:13:47

%S 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 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 9,11,14,19,29,59,15,61,31,21,8,13,33,67,34,23,35,71

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

%C Appears to contain each positive integer at least once, but only a finite number of times.

%H Amiram Eldar, <a href="/A347044/b347044.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Product_{k=floor(A001222(n)/2)+1..A001222(n)} A027746(n,k). - _Amiram Eldar_, Nov 02 2024

%e The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 7716.

%t Table[Max[Select[Divisors[n],PrimeOmega[#]==Ceiling[PrimeOmega[n]/2]&]],{n,100}]

%t 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 (Python)

%o from sympy import divisors, factorint

%o def a(n):

%o     npf = len(factorint(n, multiple=True))

%o     for d in divisors(n)[::-1]:

%o         if len(factorint(d, multiple=True)) == (npf+1)//2: return d

%o     return 1

%o print([a(n) for n in range(1, 72)]) # _Michael S. Branicky_, Aug 18 2021

%o (Python 3.8+)

%o from math import prod

%o from sympy import factorint

%o def A347044(n):

%o     fs = factorint(n,multiple=True)

%o     l = len(fs)

%o     return prod(fs[l//2:]) # _Chai Wah Wu_, Aug 20 2021

%Y The greatest divisor without the condition is A006530 (smallest: A020639).

%Y Divisors of this type are counted by A096825 (exact: A345957).

%Y The case of powers of 2 is A163403.

%Y The smallest divisor of this type is given by A347043 (exact: A347045).

%Y The exact version is A347046.

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

%Y A001222 counts all prime factors (also called bigomega).

%Y A038548 counts inferior (or superior) divisors (strict: A056924).

%Y A056239 adds up prime indices, row sums of A112798.

%Y A207375 lists central divisors (min: A033676, max: A033677).

%Y A340387 lists numbers whose sum of prime indices is twice bigomega.

%Y A340609 lists numbers whose maximum prime index divides bigomega.

%Y A340610 lists numbers whose maximum prime index is divisible by bigomega.

%Y A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).

%Y Cf. A000720, A027746, A060775, A106529, A217581, A244990, A335433, A335448.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 16 2021