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A381952 a(n) is the greatest common divisor of n and the maximum exponent in the prime factorization of n.

%I A381952 #9 Mar 12 2025 08:24:23
%S A381952 1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,1,1,3,2,1,1,1,1,1,1,
%T A381952 1,2,1,1,1,1,1,1,1,2,1,1,1,4,1,2,1,2,1,3,1,1,1,1,1,2,1,1,1,2,1,1,1,2,
%U A381952 1,1,1,3,1,1,1,2,1,1,1,4,1,1,1,2,1,1,1
%N A381952 a(n) is the greatest common divisor of n and the maximum exponent in the prime factorization of n.
%C A381952 The asymptotic density d(k) of the occurrences of each integer k >= 1 in this sequence can be calculated. E.g., d(1) = 0.75001550800919720296... (1 - the density of A368715), and d(2) = 0.16205634516436945215... (see A381953). From these densities the asymptotic mean of this sequence can be evaluated by Sum_{k>=1} k*d(k), but it seems that the expressions for d(k) for large values of k may be complicated.
%C A381952 The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 135, 1396, 14014, 140241, 1402521, 14025251, 140252636, 1402526282, 14025262924, ... . Apparently, the asymptotic mean of this sequence equals 1.402526... .
%H A381952 Amiram Eldar, <a href="/A381952/b381952.txt">Table of n, a(n) for n = 1..10000</a>
%F A381952 a(n) = gcd(n, A051903(n)).
%F A381952 a(n) >= 2 if and only if n is in A368715.
%F A381952 a(A381953(n)) = 2.
%F A381952 a(A336064(n)) = A051903(A336064(n)).
%e A381952 a(1) = gcd(1, A051903(1)) = gcd(1, 0) = 1.
%e A381952 a(4) = gcd(4, A051903(4)) = gcd(4, 2) = 2.
%e A381952 a(16) = gcd(16, A051903(16)) = gcd(16, 4) = 4.
%t A381952 a[n_] := GCD[n, If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]]; Array[a, 100]
%o A381952 (PARI) a(n) = gcd(n, if(n > 1, vecmax(factor(n)[, 2]), 0));
%Y A381952 Cf. A051903, A336064, A368715, A381953.
%K A381952 nonn,easy
%O A381952 1,4
%A A381952 _Amiram Eldar_, Mar 11 2025