A381954 The maximum exponent in the prime factorization of n that is coprime to n, or 0 if no such exponent exists.

0, 1, 1, 0, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Mar 11 2025

Keywords

• nonn
• easy

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 118, 1250, 12648, 126955, 1271426, 12720020, 127218134, 1272236949, 12722547575, .... . Apparently, the asymptotic mean of this sequence equals 1.2722... .

Examples

			a(2) = 1 since 2 = 2^1 and 1 is coprime to 2.
a(4) = 0 since 4 = 2^2 and the exponent 2 is not coprime to 4.
a(12) = 1 since 12 = 2^2 * 3^1, the exponent 2 is not coprime to 12, and the exponent 1 is coprime to 12.
		

Links

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

Crossrefs

Cf. A051903, A381952.

Programs

Mathematica

a[n_] := If[(s = Select[FactorInteger[n][[;; , 2]], CoprimeQ[#, n] &]) != {}, Max[s], 0]; a[1] = 0; Array[a, 100]

PARI

a(n) = {my(e = factor(n)[, 2], s = select(x -> if(gcd(x, n) == 1, x, 0), e)); if(#s == 0, 0, vecmax(s));}