A348986 Denominator of ratio sigma(n) / A325973(n), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

1, 1, 1, 4, 1, 1, 1, 2, 7, 1, 1, 4, 1, 1, 1, 10, 1, 7, 1, 4, 1, 1, 1, 2, 16, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 31, 1, 1, 1, 2, 1, 1, 1, 4, 7, 1, 1, 10, 29, 16, 1, 4, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 7, 34, 1, 1, 1, 4, 1, 1, 1, 17, 1, 1, 16, 4, 1, 1, 1, 10, 43, 1, 1, 4, 1, 1, 1, 2, 1, 7, 1, 4, 1, 1, 1, 2, 1, 29, 7, 74, 1, 1, 1, 2, 1

Offset: 1

Author

Antti Karttunen, Nov 06 2021

Keywords

• nonn
• frac

Comments

This is not multiplicative: a(4) = 4 and a(9) = 7, but a(36) = 31, not 28.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A048250, A034448, A325973, A325974, A348984, A348985 (numerators).

Cf. also A348947.

Programs

Mathematica

f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := Denominator[DivisorSigma[1, n]/s[n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

PARI

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A348986(n) = { my(am=A325973(n)); (am/gcd(sigma(n),am)); };

Formula

a(n) = A325973(n) / A348984(n) = A325973(n) / gcd(A000203(n), A325973(n)).