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A348735 Denominator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.

%I A348735 #13 Nov 05 2021 21:12:36
%S A348735 1,1,1,5,1,1,1,1,5,1,1,5,1,1,1,17,1,5,1,5,1,1,1,1,13,1,7,5,1,1,1,11,1,
%T A348735 1,1,25,1,1,1,1,1,1,1,5,5,1,1,17,25,13,1,5,1,7,1,1,1,1,1,5,1,1,5,65,1,
%U A348735 1,1,5,1,1,1,5,1,1,13,5,1,1,1,17,41,1,1,5,1,1,1,1,1,5,1,5,1,1,1,11,1,25,5,65
%N A348735 Denominator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.
%C A348735 This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 1444 = 2^2 * 19^2, where a(1444) = 181 <> 5*181 = a(4)*a(361). See A348740 for the list of such positions.
%H A348735 Antti Karttunen, <a href="/A348735/b348735.txt">Table of n, a(n) for n = 1..21125</a>
%F A348735 a(n) = A034448(n) / A348733(n) = A034448(n) / gcd(A003959(n), A034448(n)).
%t A348735 f[p_, e_] := (p + 1)^e/(p^e + 1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)
%o A348735 (PARI) A348735(n) = { my(f = factor(n)); denominator(prod(k=1, #f~, ((1+f[k, 1])^f[k, 2])/(1+(f[k, 1]^f[k, 2])))); };
%o A348735 (PARI)
%o A348735 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348735 A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
%o A348735 A348735(n) = { my(u=A034448(n)); (u/gcd(u, A003959(n))); };
%Y A348735 Cf. A003959, A034448, A348733, A348734 (numerators), A348740.
%K A348735 nonn,frac
%O A348735 1,4
%A A348735 _Antti Karttunen_, Nov 05 2021