A229088 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.

1, 4, 40, 224, 360, 2016, 47616, 174592, 293760, 524160, 1571328, 1782144, 3485664, 134209536, 282977280, 492101632, 746444160, 1459956960, 1684126080, 1716728832, 4428914688, 27298252800, 41233360896, 376591138560, 719045268480, 1622308746240

Offset: 1

Author

Jaroslav Krizek, Oct 24 2013

Keywords

• nonn

Comments

Numbers k such that A229087(k) = A000203(k) mod k - A024816(k) mod k = A054024(k) - A229110(k) = 0.

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.

719045268480 and 1622308746240 are also terms. - Donovan Johnson, Oct 25 2013

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

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

Conjecture: For k>1, numbers k such that GCD(sigma(k), k) = n/4. - Jaroslav Krizek, Sep 23 2014

Examples

			40 is in sequence because sigma(40) mod 40 = 90 mod 40 = antisigma(40) mod 40 = 730 mod 40 = 10.
		

Crossrefs

Cf. A000203 (sigma(n)), A024816 (antisigma(n)), A229110 (antisigma(n) mod n), A054024 (sigma(n) mod n).

Programs

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 */

Extensions

a(8)-a(23) from Donovan Johnson, Oct 24 2013

a(24)-a(26) from Jud McCranie, Oct 10 2023