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A348984 a(n) = gcd(sigma(n), A325973(n)), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

%I A348984 #14 Nov 12 2021 16:16:51
%S A348984 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 A348984 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 A348984 24,72,24,80,90,60,24,62,96,8,1,84,144,68,18,96,144,72,3,74,114,4,20,96,168,80,6,1,126,84,32,108
%N A348984 a(n) = gcd(sigma(n), A325973(n)), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.
%C A348984 This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 4, although a(4) = 1 and a(27) = 8.
%H A348984 Antti Karttunen, <a href="/A348984/b348984.txt">Table of n, a(n) for n = 1..20000</a>
%H A348984 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348984 a(n) = gcd(A000203(n), A325973(n)).
%F A348984 a(n) = gcd(A000203(n), A325974(n)) = gcd(A325973(n), A325974(n)).
%F A348984 a(n) = A000203(n) / A348985(n) = A325973(n) / A348986(n).
%t A348984 f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := GCD[s[n], DivisorSigma[1, n]]; Array[a, 100] (* _Amiram Eldar_, Nov 06 2021 *)
%o A348984 (PARI)
%o A348984 A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
%o A348984 A348984(n) = gcd(sigma(n), A325973(n));
%Y A348984 Cf. A000203, A048250, A034448, A325973, A325974, A348985, A348986.
%Y A348984 Differs from A348047 for the first time at n=108, where a(108) = 4, while A348047(108) = 8.
%Y A348984 Cf. also A348733, A348946.
%K A348984 nonn
%O A348984 1,2
%A A348984 _Antti Karttunen_, Nov 06 2021