cp's OEIS Frontend

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

%I A240923 #31 Aug 31 2025 10:45:13
%S A240923 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,
%T A240923 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,
%U A240923 0,9,8,24,0,5,0,19,0,15,0,14,0,3,0,21,0
%N A240923 a(n) = numerator(sigma(n)/n) - sigma(denominator(sigma(n)/n)).
%C A240923 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.
%C A240923 It appears that a(n) is never negative.
%C A240923 a(n) = 0 if and only if n is in A014567 (n and sigma(n) are relatively prime).
%C A240923 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
%H A240923 Reinhard Zumkeller, <a href="/A240923/b240923.txt">Table of n, a(n) for n = 1..10000</a>
%H A240923 William G. Stanton and Judy A. Holdener, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Holdener/holdener7.html">Abundancy "Outlaws" of the Form (sigma(N) + t)/N</a>, Journal of Integer Sequences , Vol 10 (2007) , Article 07.9.6.
%F A240923 a(n) = A017665(n) - sigma(A017666(n)) = sigma(n)/A009194(n) - sigma(n/A009194(n)). - _Antti Karttunen_, Aug 30 2025
%e A240923 For n=10, sigma(10)/10 = 18/10 = 9/5 = (sigma(5) + 3)/5, hence a(10)=3.
%p A240923 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
%t A240923 Table[Numerator[DivisorSigma[1, n]/n] - DivisorSigma[1, Denominator[ DivisorSigma[1, n]/n]], {n, 100}] (* _Wesley Ivan Hurt_, Aug 06 2014 *)
%o A240923 (PARI) a(n) = my(ab = sigma(n)/n); numerator(ab) - sigma(denominator(ab));
%o A240923 (PARI) A240923(n) = { my(s=sigma(n), g=gcd(s,n)); ((s/g) - sigma(n/g)); }; \\ _Antti Karttunen_, Aug 30 2025
%o A240923 (Haskell)
%o A240923 import Data.Ratio ((%), numerator, denominator)
%o A240923 a240923 n = numerator sq - a000203 (denominator sq)
%o A240923    where sq = a000203 n % n
%o A240923 -- _Reinhard Zumkeller_, Aug 05 2014
%o A240923 (Python)
%o A240923 from gmpy2 import mpq
%o A240923 from sympy import divisors
%o A240923 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
%Y A240923 Cf. A000203, A009194, A017665, A017666, A240990, A272008.
%Y A240923 Cf. A014567 (positions of 0's), A240991 (positions of 1's).
%K A240923 nonn,changed
%O A240923 1,10
%A A240923 _Michel Marcus_, Aug 03 2014