cp's OEIS Frontend

A336722 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).

%I A336722 #30 Dec 21 2024 15:20:24
%S A336722 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 A336722 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 A336722 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 A336722 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 A336722 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 A336722 Antti Karttunen, <a href="/A336722/b336722.txt">Table of n, a(n) for n = 1..65537</a>
%F A336722 a(p) = 1 for p = primes (A000040).
%F A336722 a(n) = gcd(A007955(n), A009205(n)). - _Antti Karttunen_, Aug 10 2020
%e A336722 a(6) = gcd(tau(6), sigma(6), pod(6)) = gcd(4, 12, 36) = 4.
%t A336722 a[n_] := GCD @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 100] (* _Amiram Eldar_, Aug 01 2020 *)
%o A336722 (Magma) [GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];
%o A336722 (PARI)
%o A336722 A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From A007955
%o A336722 A336722(n) = gcd(A007955(n), gcd(numdiv(n), sigma(n))); \\ _Antti Karttunen_, Aug 10 2020
%Y A336722 Cf. A009205 (gcd(tau(n), sigma(n))), A306671 (gcd(tau(n), pod(n))), A306682 (gcd(sigma(n), pod(n))).
%Y A336722 Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336723 (lcm(tau(n), sigma(n), pod(n))).
%Y A336722 Cf. A277521 (numbers k such that a(k) = tau(k) and simultaneously A336723(k) = pod(k)).
%K A336722 nonn
%O A336722 1,6
%A A336722 _Jaroslav Krizek_, Aug 01 2020
%E A336722 Data section extended up to a(105) by _Antti Karttunen_, Aug 10 2020