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A348047 a(n) = gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

%I A348047 #12 Oct 29 2021 07:55:26
%S A348047 1,3,4,1,6,12,8,3,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,12,1,42,8,8,
%T A348047 30,72,32,9,48,54,48,1,38,60,56,18,42,96,44,12,6,72,48,4,1,3,72,14,54,
%U A348047 24,72,24,80,90,60,24,62,96,8,1,84,144,68,18,96,144,72,3,74,114,4,20,96,168,80,6,1,126,84,32
%N A348047 a(n) = gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.
%C A348047 This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 196 = 4*49, where a(196) = 3, although a(4) = 1 and a(49) = 4.
%H A348047 Antti Karttunen, <a href="/A348047/b348047.txt">Table of n, a(n) for n = 1..20000</a>
%H A348047 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348047 a(n) = gcd(A000203(n), A003959(n)).
%F A348047 a(n) = gcd(A000203(n), A348029(n)) = gcd(A003959(n), A348029(n)).
%F A348047 a(n) = A000203(n)/ A348048(n) = A003959(n) / A348049(n).
%t A348047 f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[Times @@ f @@@ FactorInteger[n], DivisorSigma[1, n]]; Array[a, 100] (* _Amiram Eldar_, Oct 21 2021 *)
%o A348047 (PARI)
%o A348047 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348047 A348047(n) = gcd(sigma(n), A003959(n));
%Y A348047 Cf. A000203, A003959, A348029, A348048, A348049.
%Y A348047 Cf. also A344695.
%K A348047 nonn
%O A348047 1,2
%A A348047 _Antti Karttunen_, Oct 21 2021