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Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.
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%I #20 Nov 02 2024 09:13:41

%S 1,1,1,2,1,2,1,1,3,2,1,1,1,2,3,4,1,1,1,1,3,2,1,4,5,2,1,1,1,1,1,1,3,2,

%T 5,4,1,2,3,4,1,1,1,1,1,2,1,1,7,1,3,1,1,6,5,4,3,2,1,4,1,2,1,8,5,1,1,1,

%U 3,1,1,1,1,2,1,1,7,1,1,1,9,2,1,4,5,2,3

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

%H Amiram Eldar, <a href="/A347045/b347045.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Product_{k=1..A001222(n)/2} A027746(n,k) if A001222(n) is even, and 1 otherwise. - _Amiram Eldar_, Nov 02 2024

%e The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 6.

%t Table[If[#=={},1,Min[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]

%t a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[1 ;; np/2]]]]; 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:-1]:

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

%o return 1

%o print([a(n) for n in range(1, 88)]) # _Michael S. Branicky_, Aug 18 2021

%o (Python 3.8+)

%o from math import prod

%o from sympy import factorint

%o def A347045(n):

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

%o q, r = divmod(len(fs),2)

%o return 1 if r else prod(fs[:q]) # _Chai Wah Wu_, Aug 20 2021

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

%Y Positions of 1's are A026424.

%Y Positions of even terms are A063745 = 2*A026424.

%Y The case of powers of 2 is A072345.

%Y Positions of 2's are A100484.

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

%Y The rounded version is A347043.

%Y The greatest divisor of this type is A347046 (rounded: A347044).

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

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

%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, A001747, A027746, A038548, A056924, A063962, A106529, A140271, A244991, A324522, A335433, A335448.

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 16 2021