login
A348735
Denominator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.
6
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
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
KEYWORD
nonn,frac
AUTHOR
Antti Karttunen, Nov 05 2021
STATUS
approved