A295666 a(n) = Product_{d|n, gcd(d,n/d) is prime} gcd(d,n/d).

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 5, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 1, 16, 1, 1, 1, 4, 9, 1, 1, 16, 7, 25, 1, 4, 1, 81, 1, 16, 1, 1, 1, 16, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 144, 1, 1, 25, 4, 1, 1, 1, 16, 9, 1, 1, 16, 1, 1, 1, 16, 1, 81, 1, 4, 1, 1, 1, 16, 1, 49, 9, 100, 1, 1, 1, 16, 1

Offset: 1

Author

Antti Karttunen, Nov 26 2017

Keywords

• nonn

Examples

			For n = 12, with divisors 1, 2, 3, 4, 6, 12, we select from the sequence gcd(1,12/1), gcd(2,12/2), gcd(3,12/3), gcd(4,12/4), gcd(6,12/6), gcd(12,12/12) = 1, 2, 1, 1, 2, 1 only those that are primes, namely the two 2's, and form their product, thus a(12) = 2*2 = 4.
For n = 100, with divisors 1, 2, 4, 5, 10, 20, 25, 50, 100, we select from the sequence gcd(1,100/1), gcd(2,100/2), gcd(4,100/4), gcd(5,100/5), gcd(10,100/10), gcd(20,100/20), gcd(25,100/25), gcd(50,100/50), gcd(100,100/100) = 1, 2, 1, 5, 10, 5, 1, 2, 1, only those that are primes, namely 2, 5, 5 and 2, thus a(100) = 2*5*5*2 = 100.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003557, A056170, A010051, A294876, A294895, A295665.

Programs

Mathematica

a[n_]:=Product[GCD[i,n/i]^Boole[PrimeQ[GCD[i,n/i]]],{i,Divisors[n]}]; Array[a,105] (* Stefano Spezia, Feb 20 2024 *)

PARI

A295666(n) = { my(m=1,p); fordiv(n, d, p = gcd(d, n/d); if(isprime(p), m *= p)); m; };

Formula

a(n) = Product_{d|n} gcd(d,n/d)^A010051(gcd(d,n/d)).

a(n) = A295665(A294876(n)).

Other identities. For all n >= 1:

A001221(a(n)) = A056170(n) = A001221(A003557(n)).