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A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

%I A355934 #12 Jul 23 2022 09:56:18
%S A355934 1,3,2,7,3,1,2,3,13,9,6,14,7,1,1,31,9,39,5,21,4,9,4,1,31,7,10,14,15,3,
%T A355934 16,9,4,27,1,7,19,5,14,9,21,1,11,6,39,3,8,62,3,31,3,49,9,5,9,1,5,45,
%U A355934 30,7,31,12,26,127,7,3,17,63,8,3,36,39,37,19,62,35,4,7,20,93,11,63,14,28,27,11,5,9,45,117
%N A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.
%C A355934 Denominator of ratio A003973(n) / A000203(n). See comments in A355933.
%H A355934 Antti Karttunen, <a href="/A355934/b355934.txt">Table of n, a(n) for n = 1..20000</a>
%H A355934 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A355934 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A355934 a(n) = A000203(n) / A355932(n) = A000203(n) / gcd(A000203(n), A003973(n)).
%t A355934 f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* _Amiram Eldar_, Jul 22 2022 *)
%o A355934 (PARI)
%o A355934 A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
%o A355934 A355934(n) = { my(u=sigma(n)); (u/gcd(A003973(n), u)); };
%Y A355934 Cf. A000203, A003961, A003973, A355932, A355933 (numerators), A355940, A355941 (positions of 1's).
%Y A355934 Cf. also A336849, A349162.
%K A355934 nonn,frac
%O A355934 1,2
%A A355934 _Antti Karttunen_, Jul 22 2022