A348948 a(n) = sigma(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 20, 1, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 20, 1, 5, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 31, 121, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 217

Offset: 1

Author

Antti Karttunen, Nov 05 2021

Keywords

• nonn
• frac

Comments

Denominator of ratio A348944(n) / A000203(n).

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 91 <> 1 = a(4)*a(9).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348947 (numerators).

Programs

Mathematica

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, (Times @@ f2 @@@ f + Times @@ f3 @@@ f) / 2]; 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])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348948(n) = { my(s=sigma(n)); (s/gcd(s,A348944(n))); };

Formula

a(n) = A000203(n) / A348946(n) = A000203(n) / gcd(A000203(n), A348944(n)).