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

1,4

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) = 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

Formula

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

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 *)

Prog

(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])))); };

Crossrefs

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

Sequence in context: A087968 A340365 A110483 * A010164 A006084 A059928

Adjacent sequences: A348731 A348732 A348733 * A348735 A348736 A348737

Keyword

nonn,frac

Author

Antti Karttunen, Nov 05 2021

Status

approved