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

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, 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, 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

Offset: 1

Author

Antti Karttunen, Nov 05 2021

Keywords

• nonn
• frac

Comments

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..21125

Crossrefs

Cf. A003959, A034448, A348733, A348735 (denominators), A348740.

Programs

Mathematica

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 *)

PARI

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348734(n) = { my(u=A003959(n)); (u/gcd(u, A034448(n))); };

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])))); };

Formula

a(n) = A003959(n) / gcd(A003959(n), A034448(n)).