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A348734 Numerator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.

%I A348734 #14 Nov 05 2021 21:08:42
%S A348734 1,1,1,9,1,1,1,3,8,1,1,9,1,1,1,81,1,8,1,9,1,1,1,3,18,1,16,9,1,1,1,81,
%T A348734 1,1,1,72,1,1,1,3,1,1,1,9,8,1,1,81,32,18,1,9,1,16,1,3,1,1,1,9,1,1,8,
%U A348734 729,1,1,1,9,1,1,1,24,1,1,18,9,1,1,1,81,128,1,1,9,1,1,1,3,1,8,1,9,1,1,1,81,1,32,8
%N A348734 Numerator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.
%C A348734 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) = 360 != 1800 = 9*200 = a(4)*a(361). See A348740 for the list of such positions.
%H A348734 Antti Karttunen, <a href="/A348734/b348734.txt">Table of n, a(n) for n = 1..21125</a>
%F A348734 a(n) = A003959(n) / gcd(A003959(n), A034448(n)).
%t A348734 f[p_, e_] := (p + 1)^e/(p^e + 1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)
%o A348734 (PARI)
%o A348734 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348734 A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A348734 A348734(n) = { my(u=A003959(n)); (u/gcd(u, A034448(n))); };
%o A348734 (PARI) A348734(n) = { my(f = factor(n)); numerator(prod(k=1, #f~, ((1+f[k, 1])^f[k, 2])/(1+(f[k, 1]^f[k, 2])))); };
%Y A348734 Cf. A003959, A034448, A348733, A348735 (denominators), A348740.
%K A348734 nonn,frac
%O A348734 1,4
%A A348734 _Antti Karttunen_, Nov 05 2021