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

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, 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, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Mar 11 2025

Keywords

• nonn
• easy

Comments

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.

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... .

Examples

			a(1) = gcd(1, A051903(1)) = gcd(1, 0) = 1.
a(4) = gcd(4, A051903(4)) = gcd(4, 2) = 2.
a(16) = gcd(16, A051903(16)) = gcd(16, 4) = 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A051903, A336064, A368715, A381953.

Programs

Mathematica

a[n_] := GCD[n, If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]]; Array[a, 100]

PARI

a(n) = gcd(n, if(n > 1, vecmax(factor(n)[, 2]), 0));

Formula

a(n) = gcd(n, A051903(n)).

a(n) >= 2 if and only if n is in A368715.

a(A381953(n)) = 2.

a(A336064(n)) = A051903(A336064(n)).