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A348503 a(n) = gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

%I A348503 #11 Feb 10 2022 19:14:02
%S A348503 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 A348503 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 A348503 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 A348503 a(n) = gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.
%C A348503 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 A348503 Antti Karttunen, <a href="/A348503/b348503.txt">Table of n, a(n) for n = 1..20000</a>
%H A348503 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348503 a(n) = gcd(A000203(n), A034448(n)).
%F A348503 a(n) = gcd(A000203(n), A048146(n)) = gcd(A034448(n), A048146(n)).
%F A348503 a(n) = A000203(n) / A348504(n) = A034448(n) / A348505(n).
%t A348503 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 A348503 (PARI)
%o A348503 A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A348503 A348503(n) = gcd(sigma(n), A034448(n));
%Y A348503 Cf. A000203, A034448, A048146, A348504, A348505.
%Y A348503 Differs from A344695 for the first time at n=72, where a(72) = 15, while A344695(72) = 3.
%Y A348503 Differs from A348047 for the first time at n=27, where a(27) = 4, while A348047(27) = 8.
%K A348503 nonn
%O A348503 1,2
%A A348503 _Antti Karttunen_, Oct 29 2021