A158378 a(1) = 0, a(n) = gcd(A051904(n), A051903(n)) for n >= 2.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Author

Jaroslav Krizek, Mar 17 2009

Keywords

• nonn

Comments

a(n) for n >= 2 equals GCD of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 it deviates from A052409(n), first different term is a(10800) = a(2^4*3^3*5^2), a(10800) = gcd(2,4) = 2, A052409(10800) = gcd(2,3,4) = 1.

Examples

			For n = 12 = 2^2 * 3^1 we have a(12) = gcd(2,1) = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65536
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A157754, A051904, A051903, A052409.

Programs

Mathematica

Table[GCD @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

PARI

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A051904(n) = if((1==n),0,vecmin(factor(n)[, 2]));
A158378(n) = gcd(A051903(n),A051904(n)); \\ Antti Karttunen, Jul 12 2017

PARI

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); gcd(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

Formula

For n >= 2 holds: a(n)*A157754(n) = A051904(n)*A051903(n).

a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Sep 11 2024