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Least nonnegative integer k such that (gpf(n)!)^k is divisible by n, where gpf(n) is the greatest prime factor of n.
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%I #8 Mar 22 2024 09:17:11

%S 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,1,1,1,1,3,2,1,3,1,1,1,1,5,1,1,

%T 1,2,1,1,1,1,1,1,1,1,2,1,1,4,2,2,1,1,1,3,1,1,1,1,1,1,1,1,1,6,1,1,1,1,

%U 1,1,1,3,1,1,2,1,1,1,1,2,4,1,1,1,1,1,1

%N Least nonnegative integer k such that (gpf(n)!)^k is divisible by n, where gpf(n) is the greatest prime factor of n.

%C First differs from A088388 at n = 40.

%F a(n) > 1 if and only if n is in A057109.

%F a(n) <= A051903(n).

%F a(n) = ceiling(A371148(n)/A371149(n)). - _Pontus von Brömssen_, Mar 16 2024

%e For n = 12, gpf(n)! = 3! = 6 is not divisible by 12, but (3!)^2 = 36 is divisible by 12, so a(12) = 2.

%o (Python)

%o from sympy import factorint

%o def A362333(n):

%o f = factorint(n)

%o gpf = max(f,default=None)

%o a = 0

%o for p in f:

%o m = gpf

%o v = 0

%o while m >= p:

%o m //= p

%o v += m

%o a = max(a,-(-f[p]//v))

%o return a

%Y Cf. A006530, A051903, A057109, A088388, A371148, A371149, A371151, A371152.

%K nonn

%O 1,4

%A _Pontus von Brömssen_, Apr 16 2023