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

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, 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, 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

Offset

1,4

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) = 181 <> 5*181 = a(4)*a(361). See A348740 for the list of such positions.

Links

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

Formula

a(n) = A034448(n) / A348733(n) = A034448(n) / gcd(A003959(n), A034448(n)).

Mathematica

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 *)

Prog

(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])))); };
(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])); };
A348735(n) = { my(u=A034448(n)); (u/gcd(u, A003959(n))); };

Crossrefs

Cf. A003959, A034448, A348733, A348734 (numerators), A348740.

Sequence in context: A060904 A351084 A135469 * A170817 A170818 A046622

Adjacent sequences: A348732 A348733 A348734 * A348736 A348737 A348738

Keyword

nonn,frac

Author

Antti Karttunen, Nov 05 2021

Status

approved