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%I #10 Feb 11 2022 11:24:02
%S 1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,31,1,1,1,1,1,1,1,5,1,1,20,1,1,1,1,21,1,
%T 1,1,91,1,1,1,5,1,1,1,1,1,1,1,31,1,1,1,1,1,20,1,5,1,1,1,1,1,1,1,127,1,
%U 1,1,1,1,1,1,65,1,1,1,1,1,1,1,31,121,1,1,1,1,1,1,5,1,1,1,1,1,1,1,21,1,1,1,217
%N a(n) = sigma(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.
%C Denominator of ratio A348944(n) / A000203(n).
%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 36 = 2^2 * 3^2, where a(36) = 91 <> 1 = a(4)*a(9).
%H Antti Karttunen, <a href="/A348948/b348948.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = A000203(n) / A348946(n) = A000203(n) / gcd(A000203(n), A348944(n)).
%t f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, (Times @@ f2 @@@ f + Times @@ f3 @@@ f) / 2]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)
%o (PARI)
%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
%o A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
%o A348948(n) = { my(s=sigma(n)); (s/gcd(s,A348944(n))); };
%Y Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348947 (numerators).
%K nonn,frac
%O 1,8
%A _Antti Karttunen_, Nov 05 2021
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