A389447 - OEIS

A389447

Numerator of ratio sigma(n) / A057723(n), where A057723 is sum of positive divisors of n that are divisible by every prime that divides n and sigma is the sum of divisors function.

1, 3, 4, 7, 6, 2, 8, 15, 13, 9, 12, 14, 14, 12, 8, 31, 18, 13, 20, 7, 32, 18, 24, 10, 31, 21, 40, 4, 30, 12, 32, 63, 16, 27, 48, 91, 38, 30, 56, 9, 42, 16, 44, 14, 13, 36, 48, 62, 57, 31, 24, 49, 54, 20, 72, 60, 80, 45, 60, 28, 62, 48, 26, 127, 84, 24, 68, 21, 32, 72, 72, 65, 74, 57, 62, 70, 96, 28, 80, 31, 121

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OFFSET

1,2

COMMENTS

The sequence of fractions r(n) = A000203(n)/A057723(n), in their reduced form, A389447(n)/A389448(n), begins as: 1, 3/2, 4/3, 7/6, 6/5, 2, 8/7, 15/14, 13/12, 9/5, 12/11, 14/9, 14/13, 12/7, 8/5, 31/30, 18/17, 13/8, 20/19, 7/5, 32/21, 18/11, 24/23, 10/7, etc. As both A000203 and A057723 are multiplicative sequences, r(n) is also. Thus, if gcd(x,y)=1, then r(x*y) = r(x)*r(y), as for example, r(12) = r(3)*r(4) = 4/3 * 7/6 = 14/9. However, as an integer sequence this is not multiplicative, as for example, a(2) = 3, a(3) = 4, but a(6) = 2 <> 3*4.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to sigma(n).

FORMULA

a(n) = A000203(n) / A389446(n).

MATHEMATICA

A389447[n_] := Numerator[DivisorSigma[1, n]/(DivisorSigma[1, n/#]*#)] & [Times @@ FactorInteger[n][[All, 1]]];

Array[A389447, 100] (* Paolo Xausa, Oct 07 2025 *)

PROG

(PARI)

A057723(n) = { my(f=factor(n)); prod(i=1, #f~, sigma(f[i, 1]^f[i, 2])-1); };

A389447(n) = numerator(sigma(n)/A057723(n));