A045770 -id:A045770 - cp's OEIS frontend

A054024 Sum of the divisors of n reduced modulo n.

0, 1, 1, 3, 1, 0, 1, 7, 4, 8, 1, 4, 1, 10, 9, 15, 1, 3, 1, 2, 11, 14, 1, 12, 6, 16, 13, 0, 1, 12, 1, 31, 15, 20, 13, 19, 1, 22, 17, 10, 1, 12, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 12, 17, 8, 23, 32, 1, 48, 1, 34, 41, 63, 19, 12, 1, 58, 27, 4, 1, 51, 1, 40, 49, 64, 19, 12, 1, 26, 40

Offset: 1

Asher Auel, Jan 19 2000

If a(n) = 0, then n is a multiply-perfect number (A007691). - Alonso del Arte, Mar 30 2014

			a(12) = 4 because sigma(12) = 28 and 28 == 4 (mod 12).
a(13) = 1 because 13 is prime.
a(14) = 10 because sigma(14) = 24 and 24 == 10 (mod 14).

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe, terms 1001..20000 from Alois P. Heinz).
Walter Nissen, Abundancy : Some Resources.

Cf. A000203 (sigma), A005114 (untouchable numbers), A007691 (positions of 0's), A045768, A045769, A088834, A045770, A076496, A159907.

a054024 n = mod (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

with(numtheory): seq(sigma(i) mod i, i=1..100);

Table[Mod[DivisorSigma[1, n], n], {n, 80}] (* Alonso del Arte, Mar 30 2014 *)

a(n)=sigma(n)%n \\ Charles R Greathouse IV, Nov 04 2014

a(n) = sigma(n) mod n.

a(p) = 1 for p prime.

A045768 Numbers k such that sigma(k) == 2 (mod k).

Original entry on oeis.org

1, 20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Dan Hoey

nonn

Equivalently, Chowla function of k is congruent to 1 (mod k).
If p=2^i-3 is prime, then 2^(i-1)*p is a term of the sequence. 650 is in the sequence, but is not of this form.
Terms k from a(2) to a(14) satisfy sigma(k) = 2*k + 2, implying that sigma(k) == 0 (mod k+1). It is not known if this holds in general, for there might be solutions of sigma(k)=3k+2 or 4k+2 or ... (Comments from Jud McCranie and Dean Hickerson, updated by Jon E. Schoenfield, Sep 25 2021 and by Max Alekseyev, May 23 2025).
k | sigma(k) produces the multiperfect numbers (A007691). It is an open question whether k | sigma(k) - 1 iff k is a prime or 1. It is not known if there exist solutions to sigma(k) = 2k+1.
Sequence also gives the nonprime solutions to sigma(k) == 0 (mod k+1), k > 1. - Benoit Cloitre, Feb 05 2002
Sequence seems to give nonprime k such that the numerator of the sum of the reciprocals of the divisors of k equals k+1 (nonprime k such that A017665(k)=k+1). - Benoit Cloitre, Apr 04 2002
For k > 1, composite numbers k such that A108775(k) = floor(sigma(k)/k) = sigma(k) mod k = A054024(k). Complement of primes (A000040) with respect to A230606. There are no numbers k > 2 such that sigma(x) = k*(x+1) has a solution. - Jaroslav Krizek, Dec 05 2013

			sigma(650) = 1302 == 2 (mod 650).

R. K. Guy, Unsolved Problems in Number Theory, B2.

Amitabha Tripathi, A note on products of primes that differ by a fixed integer, Fibonacci Quart. 48 (2010), no. 2, 144-149.

Numbers k such that A054013(k)=1.

Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496.

Do[If[Mod[DivisorSigma[1, n]-2, n]==0, Print[n]], {n, 1, 10^8}]
Join[{1}, Select[Range[8000000], Mod[DivisorSigma[1, #], #]==2 &]] (* Vincenzo Librandi, Mar 11 2014 *)

is(n)=sigma(n)%n==2 || n==1 \\ Charles R Greathouse IV, Mar 09 2014

More terms from Jud McCranie, Dec 22 1999.
a(11) from Donovan Johnson, Mar 01 2012
a(12) from Giovanni Resta, Apr 02 2014
a(13) from Jud McCranie, Jun 02 2019
Edited and a(14) from Jon E. Schoenfield confirmed by Max Alekseyev, May 23 2025

A088833 Numbers n whose abundance is 8: sigma(n) - 2n = 8.

Original entry on oeis.org

56, 368, 836, 11096, 17816, 45356, 77744, 91388, 128768, 254012, 388076, 2087936, 2291936, 13174976, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216, 775397948, 3381872252, 4856970752, 6800228816, 8589344768, 44257207676, 114141404156

Offset: 1

Labos Elemer, Oct 28 2003

nonn

A subset of A045770.
If p=2^m-9 is prime (m is in the sequence A059610) then n=2^(m-1)*p is in the sequence. See comment lines of the sequence A088831. 56, 368, 128768, 2087936 & 8589344768 are of the mentioned form. - Farideh Firoozbakht, Feb 15 2008
a(28) > 10^12. - Donovan Johnson, Dec 08 2011
a(31) > 10^13. - Giovanni Resta, Mar 29 2013
a(38) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018
Any term x of this sequence can be combined with any term y of A125247 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016

			Except first 4 terms of A045770 (1, 7, 10, and 49) are here: abundances = {-1,-6,-2,-41,8,8,8,8,8,8,8,8,8,8,8,8,8}.

Giovanni Resta and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..37 (terms a(1)-a(30) from _Giovanni Resta_)

Cf. A033880, A045668, A045669, A045770, A088831, A088832, A059610, A125247 (deficiency 8).

Do[If[DivisorSigma[1,n]==2n+8,Print[n]],{n,100000000}] (* Farideh Firoozbakht, Feb 15 2008 *)

is(n)=sigma(n)==2*n+8 \\ Charles R Greathouse IV, Feb 21 2017

a(14)-a(17) from Farideh Firoozbakht, Feb 15 2008
a(18)-a(25) from Donovan Johnson, Dec 23 2008
a(26)-a(27) from Donovan Johnson, Dec 08 2011

A067702 Numbers k such that sigma(k) == 0 (mod k+2).

Original entry on oeis.org

12, 70, 88, 180, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088

Offset: 1

Benoit Cloitre, Feb 05 2002

If 2^i-5 is prime for i > 2 then let x = (2^i-5)*2^(i-1). Then sigma(x)=2*(x+2), so x is in the sequence. There are other terms that are not of this form. - Jud McCranie, Jan 12 2019

Contains terms of A088832, terms m of A088834 with (sigma(m)-6)/m = 3, terms m of A045770 with (sigma(m)-8)/m = 4, terms m of A076496 with (sigma(m)-12)/m = 6. - Max Alekseyev, May 26 2025

			sigma(180) = 546 = 3(180+2), so 180 is in the sequence.

Contains subsequence A088832.

Cf. A045768, A156560, A059608, A045770, A076496, A088834.

Select[Range[84*10^5],Divisible[DivisorSigma[1,#],#+2]&] (* Harvey P. Dale, May 11 2018 *)

isok(n) = sigma(n) % (n+2) == 0; \\ Michel Marcus, Nov 22 2013

a(9)-a(11) from Michel Marcus, Nov 22 2013
a(12)-a(13) from Jud McCranie, Jan 12 2019
a(14) from Jud McCranie, Jan 13 2019
a(15) from Jud McCranie, Dec 02 2019
a(16) from Max Alekseyev, May 26 2025

A274566 Numbers k such that sigma(k) == 0 (mod k-10).

Original entry on oeis.org

6, 9, 11, 12, 14, 22, 40, 42, 46, 154, 190, 2656, 6490, 44650, 318250, 1360810, 1503370, 1788490, 3214090, 103712410, 3915380170, 6077111050, 9796360330, 10828121356, 33086522327050, 35966517350410, 11577093570201610, 16726040141635450, 576460762503970816

Offset: 1

Paolo P. Lava, Jul 06 2016

nonn

			sigma(11) mod (11 - 10) = 12 mod 1 = 0.

Contains subsequence A223607.
Contains 2 * odd terms of A101223.
Cf. A045770, A067702, A088833, A181598, A274551, A274552, A274553, A274554, A274556, A274557, A274558, A274559, A274560, A274561, A274562, A274563, A274564, A274565.

[n: n in [1..2*10^6] | n ne 10 and SumOfDivisors(n) mod (n-10) eq 0 ]; // Vincenzo Librandi, Jul 06 2016

k = -10; Select[Range[1, 10^7], # + k != 0 && Mod[DivisorSigma[1, #], # + k] == 0 &] (* Vincenzo Librandi, Jul 06 2016 *)

isok(k) = (k!=10) && !(Mod(sigma(k), k-10)); \\ Michel Marcus, May 30 2025

a(19)-a(24) from Giovanni Resta, Jul 06 2016
a(25)-a(26) from Jud McCranie, Dec 02 2019
Terms 6,9 inserted and a(27)-a(29) added by Max Alekseyev, May 30 2025

A076496 Numbers k such that sigma(k) == 12 (mod k).

Original entry on oeis.org

1, 6, 11, 24, 30, 42, 54, 66, 78, 102, 114, 121, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 780, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338

Offset: 1

Labos Elemer, Oct 21 2002

nonn

			6*p is a solution if p > 3 is prime, since sigma(6*p) = 1 + 2 + 3 + 6 + p + 2*p + 3*p + 6*p = 12*(p+1) = 2*6*p + 12 = 2*k + 12. These are "regular" solutions. Also k = 121, 304 are "singular" solutions. See other remainders in cross-references.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000, corrected by _Sidney Cadot_, Feb 05 2023

Cf. A054024, A045768, A045769, A045770, A076495, A088834.

Cf. A141545 (a subsequence).

Select[Range[2000], Mod[DivisorSigma[1, #] - 12, #] == 0 &] (* Vincenzo Librandi, Mar 11 2014, corrected by Amiram Eldar, Jan 04 2023 *)

isok(k) = Mod(sigma(k), k) == 12; \\ Michel Marcus, Jan 04 2023

Initial term 1 added by Vincenzo Librandi, Mar 11 2014

Terms 6 and 11 inserted by Michel Marcus, Jan 04 2023

A045769 Numbers k such that sigma(k) == 4 (mod k).

Original entry on oeis.org

1, 3, 9, 12, 70, 88, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088, 9223372026117357568

Offset: 1

Dan Hoey

Every number of the form 2^(j-1)*(2^j - 5), where 2^j - 5 is prime, is a term. See A059608. - Jon E. Schoenfield, Jun 02 2019

Contains subsequence A088832.

Cf. A054024, A045768, A059608, A088834, A045770, A076496.

isok(k) = Mod(sigma(k), k) == 4; \\ Michel Marcus, Jan 04 2023

a(13) from Harvey P. Dale, Mar 20 2011
Initial term 1 inserted and a(14)-a(16) from Donovan Johnson, Mar 01 2012
Term 3 inserted by Michel Marcus, Jan 04 2023
a(18) from Jon E. Schoenfield confirmed, and a(17), a(19) added by Max Alekseyev, Jun 08 2025

A274553 Numbers k such that sigma(k) == 0 (mod k+4).

Original entry on oeis.org

9, 56, 368, 780, 836, 2352, 11096, 17816, 45356, 77744, 91388, 128768, 254012, 388076, 430272, 2087936, 2291936, 13174976, 29465852, 35021696, 45335936, 120888092, 184773312, 260378492, 381236216, 775397948, 3381872252, 4856970752, 6800228816, 8589344768, 44257207676, 114141404156, 1461083549696, 1471763808896, 2199013818368

Offset: 1

Paolo P. Lava, Jun 28 2016

nonn

			sigma(9) mod 9+4 = 13 mod 13 = 0.

Cf. A045770, A067702, A088833, A181598, A274551, A274552, A274554-A274566.

[n: n in [1..2*10^6] | SumOfDivisors(n) mod (n+4) eq 0 ]; // Vincenzo Librandi, Jul 02 2016

k = 4; Select[Range[Abs@ k + 1, 10^6], Mod[DivisorSigma[1, #], # + k] == 0 &] (* Michael De Vlieger, Jul 01 2016 *)

a(18)-a(35) from Giovanni Resta

A274557 Numbers k such that sigma(k) == 0 (mod k+6).

Original entry on oeis.org

6, 24, 25, 30, 42, 54, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362

Offset: 1

Paolo P. Lava, Jul 05 2016

nonn

			sigma(6) mod (6+6) = 12 mod 12 = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A045770, A067702, A088833, A181598, A274551-A274556, A274558-A274566.

Select[Range@ 1400, Mod[DivisorSigma[1, #], # + 6] == 0 &] (* Michael De Vlieger, Jul 05 2016 *)

is(n)=sigma(n)%(n+6)==0 \\ Charles R Greathouse IV, Jul 16 2016

A274562 Numbers k such that sigma(k) == 0 (mod k-8).

Original entry on oeis.org

5, 6, 7, 9, 10, 11, 12, 14, 17, 38, 92, 168, 170, 248, 752, 988, 2528, 2808, 8648, 12008, 34688, 63248, 117808, 526688, 531968, 820808, 1292768, 1495688, 2095208, 2112512, 3477608, 4495808, 8419328, 12026888, 13192768, 16102808, 26347688, 29322008, 33653888, 169371008, 173631608, 293947648, 537116672, 883927808, 2147975168, 2493705728, 5556840416, 13092865928, 42783299288, 69662739968, 80999455688, 217898810368, 546409576448, 1020401174528, 1081071376208, 1282330216448, 1473186024448, 1577975316488, 1608005456768

Offset: 1

Paolo P. Lava, Jul 05 2016

nonn

			sigma(9) mod (9 - 8) = 13 mod 1 = 0.

Cf. A000203, A045770, A067702, A088833, A181598, A274551, A274552, A274553, A274554, A274556, A274557, A274558, A274559, A274560, A274561, A274563, A274564, A274565, A274566.

Select[Range[1, 10^6], # - 8 != 0 && Mod[DivisorSigma[1, #], # - 8] == 0 &] (* Michael De Vlieger, Jul 05 2016 *)

a(33)-a(59) from Giovanni Resta, Jul 05 2016

Terms 5,6,7 inserted by Max Alekseyev, Jun 04 2025

cp's OEIS Frontend

A054024 Sum of the divisors of n reduced modulo n.

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A045768 Numbers k such that sigma(k) == 2 (mod k).

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A088833 Numbers n whose abundance is 8: sigma(n) - 2n = 8.

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A067702 Numbers k such that sigma(k) == 0 (mod k+2).

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A274566 Numbers k such that sigma(k) == 0 (mod k-10).

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A076496 Numbers k such that sigma(k) == 12 (mod k).

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A045769 Numbers k such that sigma(k) == 4 (mod k).

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