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%I #24 Sep 08 2022 08:46:25
%S 1,1,1,1,1,4,1,1,1,2,1,2,1,4,1,1,1,3,1,2,1,4,1,4,1,2,1,2,1,8,1,1,1,2,
%T 1,1,1,4,1,2,1,8,1,2,3,4,1,2,1,1,1,2,1,8,1,8,1,2,1,12,1,4,1,1,1,8,1,2,
%U 1,8,1,3,1,2,1,2,1,8,1,2,1,2,1,4,1,4,1,4,1,6,1,2,1,4,1,12,1,1,3,1,1,8,1,2,1
%N a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).
%C a(n) = tau(n) for numbers n: 1, 6, 14, 22, 30, 38, 42, 46, 54, 56, 60, 62, 66, 70, 78, 86, 94, 96, 102, ...
%H Antti Karttunen, <a href="/A336722/b336722.txt">Table of n, a(n) for n = 1..65537</a>
%F a(p) = 1 for p = primes (A000040).
%F a(n) = gcd(A007955(n), A009205(n)). - _Antti Karttunen_, Aug 10 2020
%e a(6) = gcd(tau(6), sigma(6), pod(6)) = gcd(4, 12, 36) = 4.
%t a[n_] := GCD @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 100] (* _Amiram Eldar_, Aug 01 2020 *)
%o (Magma) [GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]]
%o (PARI)
%o A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From A007955
%o A336722(n) = gcd(A007955(n), gcd(numdiv(n), sigma(n))); \\ _Antti Karttunen_, Aug 10 2020
%Y Cf. A009205 (gcd(tau(n), sigma(n)), A306671 (gcd(tau(n), pod(n)), A306682 (gcd(sigma(n), pod(n)).
%Y Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336723 (lcm(tau(n), sigma(n), pod(n))).
%Y Cf. A277521 (numbers k such that a(k) = tau(k) and simultaneously A336723(k) = pod(k)).
%K nonn
%O 1,6
%A _Jaroslav Krizek_, Aug 01 2020
%E Data section extended up to a(105) by _Antti Karttunen_, Aug 10 2020
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