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

%I A295666 #16 Feb 21 2024 10:36:44
%S A295666 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,
%T A295666 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,
%U A295666 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
%N A295666 a(n) = Product_{d|n, gcd(d,n/d) is prime} gcd(d,n/d).
%H A295666 Antti Karttunen, <a href="/A295666/b295666.txt">Table of n, a(n) for n = 1..65537</a>
%F A295666 a(n) = Product_{d|n} gcd(d,n/d)^A010051(gcd(d,n/d)).
%F A295666 a(n) = A295665(A294876(n)).
%F A295666 Other identities. For all n >= 1:
%F A295666 A001221(a(n)) = A056170(n) = A001221(A003557(n)).
%e A295666 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.
%e A295666 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.
%t A295666 a[n_]:=Product[GCD[i,n/i]^Boole[PrimeQ[GCD[i,n/i]]],{i,Divisors[n]}]; Array[a,105] (* _Stefano Spezia_, Feb 20 2024 *)
%o A295666 (PARI) A295666(n) = { my(m=1,p); fordiv(n, d, p = gcd(d, n/d); if(isprime(p), m *= p)); m; };
%Y A295666 Cf. A003557, A056170, A010051, A294876, A294895, A295665.
%K A295666 nonn
%O A295666 1,4
%A A295666 _Antti Karttunen_, Nov 26 2017