A348503 - OEIS

%I #11 Feb 10 2022 19:14:02

%S 1,3,4,1,6,12,8,3,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,12,1,42,4,8,

%T 30,72,32,3,48,54,48,1,38,60,56,18,42,96,44,12,6,72,48,4,1,3,72,14,54,

%U 12,72,24,80,90,60,24,62,96,8,1,84,144,68,18,96,144,72,15,74,114,4,20,96,168,80,6,1,126,84,32

%N a(n) = gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

%C This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 15 != 3*1 = a(8)*a(9).

%H Antti Karttunen, <a href="/A348503/b348503.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = gcd(A000203(n), A034448(n)).

%F a(n) = gcd(A000203(n), A048146(n)) = gcd(A034448(n), A048146(n)).

%F a(n) = A000203(n) / A348504(n) = A034448(n) / A348505(n).

%t f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (fct = FactorInteger[n]), Times @@ f2 @@@ fct]; Array[a, 100] (* _Amiram Eldar_, Oct 29 2021 *)

%o (PARI)

%o A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448

%o A348503(n) = gcd(sigma(n), A034448(n));

%Y Cf. A000203, A034448, A048146, A348504, A348505.

%Y Differs from A344695 for the first time at n=72, where a(72) = 15, while A344695(72) = 3.

%Y Differs from A348047 for the first time at n=27, where a(27) = 4, while A348047(27) = 8.

%K nonn

%O 1,2

%A _Antti Karttunen_, Oct 29 2021