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

1,8

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)

Formula

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

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 *)

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])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348948(n) = { my(s=sigma(n)); (s/gcd(s, A348944(n))); };

Crossrefs

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

Sequence in context: A354987 A361793 A257099 * A291578 A165485 A179947

Adjacent sequences: A348945 A348946 A348947 * A348949 A348950 A348951

Keyword

nonn,frac

Author

Antti Karttunen, Nov 05 2021

Status

approved