A240923 a(n) = numerator(sigma(n)/n) - sigma(denominator(sigma(n)/n)).

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 3, 0, 4, 2, 0, 0, 1, 0, 3, 0, 6, 0, 2, 0, 7, 0, 1, 0, 6, 0, 0, 4, 9, 0, 0, 0, 10, 0, 2, 0, 8, 0, 9, 2, 12, 0, 3, 0, 0, 6, 7, 0, 7, 0, 7, 0, 15, 0, 8, 0, 16, 0, 0, 0, 12, 0, 9, 8, 24, 0, 5, 0, 19, 0, 15, 0, 14, 0, 3, 0, 21, 0

Offset

1,10

Comments

a(n) is the integer t, such that if sigma(n)/n is written in its reduced form, nk/dk = A017665(n)/A017666(n), then we have (sigma(dk)+t)/dk.

It appears that a(n) is never negative.

a(n) = 0 if and only if n is in A014567 (n and sigma(n) are relatively prime).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

William G. Stanton and Judy A. Holdener, Abundancy "Outlaws" of the Form (sigma(N) + t)/N, Journal of Integer Sequences , Vol 10 (2007) , Article 07.9.6.

Example

For n=10, sigma(10)/10 = 18/10 = 9/5 = (sigma(5) + 3)/5, hence a(10)=3.

Maple

with(numtheory): A240923:=n->numer(sigma(n)/n) - sigma(denom(sigma(n)/n)): seq(A240923(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2014

Mathematica

Table[Numerator[DivisorSigma[1, n]/n] - DivisorSigma[1, Denominator[ DivisorSigma[1, n]/n]], {n, 100}] (* Wesley Ivan Hurt, Aug 06 2014 *)

Prog

(PARI) a(n) = my(ab = sigma(n)/n); numerator(ab) - sigma(denominator(ab));
(Haskell)
import Data.Ratio ((%), numerator, denominator)
a240923 n = numerator sq - a000203 (denominator sq)
   where sq = a000203 n % n
-- Reinhard Zumkeller, Aug 05 2014
(Python)
from gmpy2 import mpq
from sympy import divisors
map(lambda x: x.numerator-sum(divisors(x.denominator)), [mpq(sum(divisors(n)), n) for n in range(1, 10**5)]) # Chai Wah Wu, Aug 05 2014

Crossrefs

Cf. A014567, A017665, A017666.

Cf. A000203.

Sequence in context: A297871 A292130 A291971 * A272727 A333791 A323135

Adjacent sequences: A240920 A240921 A240922 * A240924 A240925 A240926

Keyword

nonn

Author

Michel Marcus, Aug 03 2014

Status

approved