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%I #11 Nov 11 2021 21:38:26
%S 1,1,1,7,1,1,1,5,13,1,1,7,1,1,1,31,1,13,1,7,1,1,1,5,31,1,5,7,1,1,1,7,
%T 1,1,1,91,1,1,1,5,1,1,1,7,13,1,1,31,57,31,1,7,1,5,1,5,1,1,1,7,1,1,13,
%U 127,1,1,1,7,1,1,1,65,1,1,31,7,1,1,1,31,121,1,1,7,1,1,1,5,1,13,1,7,1,1,1,7,1,57,13
%N Numerator of ratio sigma(n) / A325973(n), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.
%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 108 = 4*27, where a(108) = 70 <> 35 = 7*5 = a(4)*(27).
%H Antti Karttunen, <a href="/A348985/b348985.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) = A000203(n) / A348984(n) = sigma(n) / gcd(sigma(n), A325973(n)).
%t 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_] := Numerator[DivisorSigma[1, n]/s[n]]; Array[a, 100] (* _Amiram Eldar_, Nov 06 2021 *)
%o (PARI)
%o A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
%o A348985(n) = { my(s=sigma(n)); (s/gcd(s, A325973(n))); };
%Y Cf. A000203, A048250, A034448, A325973, A325974, A348984, A348986 (denominators).
%Y Differs from A348048 for the first time at n=108, where a(108) = 70, while A348048(108) = 35.
%Y Cf. also A348948.
%K nonn,frac
%O 1,4
%A _Antti Karttunen_, Nov 06 2021
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