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

Author

Michel Marcus, Aug 03 2014

Keywords

• nonn
• changed

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).

That a(n) >= 0 is easily seen from the formula a(n) = sigma(n)/A009194(n) - sigma(n/A009194(n)), where A009194(n) = gcd(n,sigma(n)). - Antti Karttunen, Aug 30 2025

Examples

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

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.

Crossrefs

Cf. A000203, A009194, A017665, A017666, A240990, A272008.

Cf. A014567 (positions of 0's), A240991 (positions of 1's).

Programs

Haskell

import Data.Ratio ((%), numerator, denominator)
a240923 n = numerator sq - a000203 (denominator sq)
   where sq = a000203 n % n
-- Reinhard Zumkeller, Aug 05 2014

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 *)

PARI

a(n) = my(ab = sigma(n)/n); numerator(ab) - sigma(denominator(ab));

PARI

A240923(n) = { my(s=sigma(n), g=gcd(s,n)); ((s/g) - sigma(n/g)); }; \\ Antti Karttunen, Aug 30 2025

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

Formula

a(n) = A017665(n) - sigma(A017666(n)) = sigma(n)/A009194(n) - sigma(n/A009194(n)). - Antti Karttunen, Aug 30 2025