A054024 -id:A054024 - cp's OEIS frontend

A242486 Numbers n such that A242480(x) = n has no solution.

1, 4, 10, 14, 22, 26, 34, 36, 38, 46, 48, 50, 58, 60, 62, 74, 82, 84, 86, 90, 94, 98, 106, 108, 110, 118, 122, 130, 132, 134, 142, 146, 154, 156, 158, 166, 170, 178, 182, 190, 194, 202, 206, 210, 214, 218, 226, 230, 238, 242, 250, 252, 254, 262, 266, 270, 274

Offset: 1

Jaroslav Krizek, May 27 2014

nonn

A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

All values of a(n) are even for n > 1. Complement of A242485.

			14 is in the sequence because there is no x whose A242480(x) = 14.

Cf. A242480, A242481, A242482, A242483, A242484, A242485.

A056075 Numbers m such that m divides sigma(m) - d(m).

Original entry on oeis.org

1, 4, 56, 7192, 7232, 7912, 10792, 17272, 30592, 114256, 2154584, 3428368, 44375136, 89245784, 2739393699744, 36993585958528, 47319950478240, 118122891971648, 943226995376128, 2737657760695168, 5020331545072768, 36028789368553472, 40256362055287184, 42381542060395136, 950808877965961856, 2616769087480013696, 3515864044679266304, 4611826686121443328, 9223371897268338688

Offset: 1

Robert G. Wilson v, Jul 26 2000

nonn

Or, numbers n such that sigma(n) = k*n + d(n) for some k.
For most terms > 4, sigma(n) = 2*n + d(n), i.e., k=2. However, for the 12th term, k=3.
If p = 2^m-(2m+1) is prime and n = 2^(m-1)*p then sigma(n) = 2*n+d(n), i.e., k=2 and n is in the sequence. 56, 7232, 30592, 36028789368553472, 9223371897268338688 and 29230032746618058364071726105239688547563879792624 are such terms of the sequence. - Farideh Firoozbakht, Aug 19 2013
Note that for m = a(17) = 47319950478240, we have (sigma(m) - d(m))/m = 4. - Max Alekseyev, May 31 2025

Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13.3 (2010), 18 pp. Article ID 10.3.1.

Cf. A000203, A000005, A054024.

Do[If[Mod[DivisorSigma[1, n]-DivisorSigma[0, n], n]==0, Print[n]], {n, 1, 10^8}]

is(n)=my(f=factor(n)); (sigma(f)-numdiv(f))%n==0 \\ Charles R Greathouse IV, Nov 04 2016

Numbers n such that A000203(n) (mod n) == A000005(n) or A054024(n)=A000005(n). - Labos Elemer, Apr 12 2002

a(15) from Giovanni Resta, Nov 07 2019

a(16)-a(29) from Max Alekseyev, May 31 2025

A117553 When adding some positive divisors of n in order from lowest divisor to highest divisor, a(n) is lowest sum achievable which is >= n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 16, 14, 24, 24, 31, 18, 21, 20, 22, 32, 36, 24, 24, 31, 42, 40, 28, 30, 42, 32, 63, 48, 54, 48, 37, 38, 60, 56, 50, 42, 54, 44, 84, 78, 72, 48, 52, 57, 93, 72, 98, 54, 66, 72, 64, 80, 90, 60, 78, 62, 96, 104, 127, 84, 78, 68, 126, 96, 74, 72

Offset: 1

Leroy Quet, Mar 28 2006

nonn

Often, but not always, a(n)=n+A054024(n). The exceptions to this rule are at n=24, 36, 48, 60, 72, 84,90, 96, 108, ... - R. J. Mathar, Mar 14 2007

			12's divisors are 1,2,3,4,6 and 12. Adding the divisors in order we have:
1 = 1, 1+2 = 3, 1+2+3 = 6, 1+2+3+4 = 10, 1+2+3+4+6 = 16 and 1+2+3+4+6+12 = 28.
Of these sums, 1+2+3+4+6 = 16 is the lowest which is >= 12. So a(12) = 16.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A117552, A125746, A125747.

A117553 := proc(n) local divs,a,i ; divs := numtheory[divisors](n) ; a := op(1,divs) ; i := 1 ; while a < n do i := i+1 ; a := a+op(i,divs) ; od ; RETURN(a) ; end: for n from 1 to 80 do printf("%d, ",A117553(n)) ; od ; # R. J. Mathar, Mar 14 2007

Table[Select[Accumulate[Divisors[n]],#>=n&,1],{n,80}]//Flatten (* Harvey P. Dale, Apr 05 2017 *)

More terms from R. J. Mathar, Mar 14 2007

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.

Original entry on oeis.org

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 *)

A076495 Smallest x such that sigma(x) mod x = n, or 0 if no such x exists.

Original entry on oeis.org

2, 20, 4, 9, 0, 25, 8, 10, 15, 14, 21, 24, 27, 22, 16, 26, 39, 208, 36, 34, 51, 38, 57, 112, 95, 46, 69, 48, 115, 841, 32, 58, 45, 62, 93, 660, 155, 1369, 162, 44, 63, 1681, 50, 82, 123, 52, 129, 60, 75, 94, 72, 352, 235, 90, 329, 84, 99, 68, 265, 96, 371, 118, 64, 76

Offset: 1

Labos Elemer, Oct 21 2002

nonn

At present, the 0 entry for n=5 is only a conjecture.
For n <= 1000, a(5) and a(898) are the only terms not found using x <= 10^11. - Donovan Johnson, Sep 20 2012
10^11 < a(898) <= 140729946996736. - Donovan Johnson, Sep 28 2013
a(898) > 10^13 and the same bound holds for a(5), if it exists. - Giovanni Resta, Apr 02 2014
a(5) > 1.5*10^14, if it exists. - Jud McCranie, Jun 02 2019

			n=1: a(1) = smallest prime = 2.
n=3: a(3) = 4 since sigma(4) mod 4 = 7 mod 4 = 3.
n=5: Very difficult case (see Comments section).

Donovan Johnson, Table of n, a(n) for n = 1..1000
Carl Pomerance, On the congruences σ(n) ≡ a (mod n) and n ≡ a (mod φ(n)), Acta Arithmetica 26:3 (1974-1975), pp. 265-272. (See theorem 4.)

Cf. A045768, A045769, A045770, A054024.

f[x_] := s=Mod[DivisorSigma[1, n], n]; t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000}]; t

a(n)=my(k);while(sigma(k++)%k!=n,);k \\ Charles R Greathouse IV, Dec 28 2013

A229089 Numbers n such that 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.

Original entry on oeis.org

3, 5, 6, 7, 9, 11, 13, 14, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93

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 A229090 with respect to A000027, where
A229088 = numbers n such that sigma(n) mod n = antisigma(n) mod n,
A229090 = numbers n such that sigma(n) mod n > antisigma(n) mod n.

			Number 11 is in sequence because sigma(11) mod 11 = 12 mod 11 = 1 < antisigma(11) mod 11 = 54 mod 11 = 10.

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

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

Select[Range[100],Mod[Total[Complement[Range[#],Divisors[#]]],#]> Mod[ DivisorSigma[ 1,#],#]&] (* Harvey P. Dale, Jan 24 2022 *)

A054027 Numbers that do not divide their sum of divisors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Asher Auel, Jan 19 2000

nonn

Does not contain numbers like 1, 120, 672, 30240, 32760, 523776,.. which are in A132999. - R. J. Mathar, Jun 13 2025

Complement of A007691. Cf. A000203, A054024.

with(numtheory): [seq(`if`(sigma(i) mod i <> 0,i,print( )),i=1..90)];

Select[Range[100],!Divisible[DivisorSigma[1,#],#]&] (* Harvey P. Dale, May 29 2019 *)

isok(m) = (sigma(m) % m) != 0; \\ Michel Marcus, Jun 20 2021

A055681 Numbers k that divide sigma(k)-phi(k).

Original entry on oeis.org

1, 2, 12, 42, 1242, 75960, 1447488, 3506976, 6137440, 10834488, 17156160, 90288000, 431440416, 454460160, 704592000, 1385119360, 1588268480, 10674673152, 24913095480, 31103703540, 56015374080, 80767843200, 129631788000, 463308768000, 469897798656, 834460413696

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

a(37) > 10^13. - Giovanni Resta, Jun 12 2013

Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 10^13)

Cf. A011774, A000203, A000010, A054024, A056075.

with(numtheory): A055681:=n->`if`(sigma(n)-phi(n) mod n=0,n,NULL): seq(A055681(n), n=1..10^5); # Wesley Ivan Hurt, Sep 13 2014

Do[If[Mod[DivisorSigma[1, n]-EulerPhi[n], n]==0, Print[n]], {n, 1, 10^9}]

for(n=1,10^8,if((sigma(n)-eulerphi(n))%n==0,print1(n,", "))) \\ Derek Orr, Sep 13 2014

a(16)-a(26) from Donovan Johnson, Feb 28 2012

cp's OEIS Frontend

A242486 Numbers n such that A242480(x) = n has no solution.

Views

Author

Keywords

Comments

Examples

Crossrefs

A056075 Numbers m such that m divides sigma(m) - d(m).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A117553 When adding some positive divisors of n in order from lowest divisor to highest divisor, a(n) is lowest sum achievable which is >= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A076495 Smallest x such that sigma(x) mod x = n, or 0 if no such x exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A229089 Numbers n such that 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs