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%I #37 Oct 12 2023 03:13:10
%S 1,4,40,224,360,2016,47616,174592,293760,524160,1571328,1782144,
%T 3485664,134209536,282977280,492101632,746444160,1459956960,
%U 1684126080,1716728832,4428914688,27298252800,41233360896,376591138560,719045268480,1622308746240
%N Numbers k such that sigma(k) mod k = antisigma(k) mod k, where sigma(k) = A000203(k) = sum of divisors of k, antisigma(k) = A024816(k) = sum of non-divisors of k.
%C Numbers k such that A229087(k) = A000203(k) mod k - A024816(k) mod k = A054024(k) - A229110(k) = 0.
%C Complement of union A229089 and A229090 with respect to A000027; where A229089 = numbers k such that sigma(k) mod k < antisigma(k) mod k, A229090 = numbers k such that sigma(k) mod k > antisigma(k) mod k.
%C 719045268480 and 1622308746240 are also terms. - _Donovan Johnson_, Oct 25 2013
%C If a number m is in this sequence and k(m) = A054024(m)/m = A229110(m)/m then k(m) = 0 for odd m (for number 1 and eventually odd multiply-perfect numbers m > 1). Conjecture: k(m) = 1/4 or 3/4 for all even m. Sequence of values k(m): 0, 3/4, 1/4, 1/4, 1/4, 1/4, 3/4, 1/4, 3/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4, 1/4, 3/4, 3/4, 3/4, 3/4, 1/4, 3/4, 3/4, ... . Value k(m) = 3/4 also for m = 719045268480 and 1622308746240. - _Jaroslav Krizek_, Jun 19 2014
%C Also, the denominator of sigma(k)/k (reduced to lowest terms) of the currently known terms, except 1, are all 4: 1, 7/4, 9/4, 9/4, 13/4, 13/4, 11/4, 9/4, 15/4, 17/4, 13/4, 15/4, 15/4, 11/4, 15/4, 9/4, 19/4, 19/4, 19/4, 15/4, 13/4, 19/4, 15/4. - _Michel Marcus_, Jun 21 2014
%C Conjecture: For k>1, numbers k such that GCD(sigma(k), k) = n/4. - _Jaroslav Krizek_, Sep 23 2014
%e 40 is in sequence because sigma(40) mod 40 = 90 mod 40 = antisigma(40) mod 40 = 730 mod 40 = 10.
%o (PARI) for(n=1, 10^9, s=sigma(n); t=n*(n+1)/2; if(s%n==(t-s)%n, print1(n ", "))) /* _Donovan Johnson_, Oct 24 2013 */
%Y Cf. A000203 (sigma(n)), A024816 (antisigma(n)), A229110 (antisigma(n) mod n), A054024 (sigma(n) mod n).
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Oct 24 2013
%E a(8)-a(23) from _Donovan Johnson_, Oct 24 2013
%E a(24)-a(26) from _Jud McCranie_, Oct 10 2023
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