A229089 -id:A229089 - cp's OEIS frontend

A229087 a(n) = sigma(n) mod n - antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisor of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

0, 1, -1, 0, -3, -3, -5, 2, -1, 1, -9, 2, -11, -1, 3, 6, -15, -3, -17, -6, 1, -5, -21, 12, -13, -7, -1, -14, -27, 9, -29, 14, -3, -11, -9, -16, -35, -13, -5, 0, -39, 3, -41, 14, 21, -17, -45, -16, -33, 11, -9, 14, -51, -3, -21, -12, -11, -23, -57, 6, -59, -25

Offset: 1

Jaroslav Krizek, Oct 24 2013

sign

Sequence contains anomalous increased frequency of values 14 (see A229115), a(n) = 14 for n = 32, 44, 52, 68, 76, 92, ... ).

			For n = 32; a(32 ) = sigma(32) mod 32 - antisigma(32) mod 32 = 63 mod 32 - 465 mod 32 = 31 - 17 = 14.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)), A024816 (antisigma(n)).
Cf. A054024 (sigma(n) mod n), A229110(antisigma(n) mod n).
Cf. A229088 (numbers n such that sigma(n) mod n = antisigma(n) mod n).
Cf. A229089 (numbers n such that sigma(n) mod n < antisigma(n) mod n).
Cf. A229090 (numbers n such that sigma(n) mod n > antisigma(n) mod n).

a(n) = A000203(n) mod n - A024816(n) mod n = A054024(n) - A229110(n).

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.

Original entry on oeis.org

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

Jaroslav Krizek, Oct 24 2013

nonn

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

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

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

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 */

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

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

A229090 Numbers n such that sigma(n) mod n > antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisors of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

Original entry on oeis.org

2, 8, 10, 12, 15, 16, 21, 24, 30, 32, 42, 44, 45, 50, 52, 60, 63, 64, 68, 75, 76, 80, 92, 99, 105, 110, 116, 117, 124, 126, 128, 130, 135, 136, 140, 144, 147, 148, 150, 152, 153, 154, 160, 164, 165, 168, 170, 171, 172, 182, 184, 188, 189, 190, 195, 198, 200

Offset: 1

Jaroslav Krizek, Oct 24 2013

nonn

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

Complement of union A229088 and A229089 with respect to A000027, where A229088 = numbers n such that sigma(n) mod n = antisigma(n) mod n, A229089 = numbers n such that sigma(n) mod n < antisigma(n) mod n.

			Number 12 is in sequence because sigma(12) mod 12 = 28 mod 12 = 4 > antisigma(12) mod 12 = 50 mod 12 = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)), A024816 (antisigma(n)).

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

smQ[n_]:=Module[{sig=DivisorSigma[1,n]},Mod[sig,n]>Mod[(n(n+1))/2-sig,n]]; Select[Range[200],smQ] (* Harvey P. Dale, Dec 23 2013 *)

cp's OEIS Frontend

A229087 a(n) = sigma(n) mod n - antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisor of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

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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.

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A229090 Numbers n such that sigma(n) mod n > antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisors of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

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