A348985 Numerator 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, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 5, 7, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 7, 1, 57, 13

Offset: 1

Author

Antti Karttunen, Nov 06 2021

Keywords

• nonn
• frac

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 108 = 4*27, where a(108) = 70 <> 35 = 7*5 = a(4)*(27).

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, A348986 (denominators).

Differs from A348048 for the first time at n=108, where a(108) = 70, while A348048(108) = 35.

Cf. also A348948.

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_] := Numerator[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))));
A348985(n) = { my(s=sigma(n)); (s/gcd(s, A325973(n))); };

Formula

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