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

1,2

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

Links

Table of n, a(n) for n=1..26.

Example

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

Prog

(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 */

Crossrefs

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

Sequence in context: A271286 A174644 A273310 * A270088 A115286 A119635

Adjacent sequences: A229085 A229086 A229087 * A229089 A229090 A229091

Keyword

nonn

Author

Jaroslav Krizek, Oct 24 2013

Extensions

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

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

Status

approved