A274556 -id:A274556 - cp's OEIS frontend

A274551 Numbers k such that sigma(k) == 0 (mod k+3).

4, 8925, 32445, 442365

Offset: 1

Paolo P. Lava, Jun 28 2016

a(5) > 10^8 if it exists. - Felix Fröhlich, Jul 01 2016
No more terms < 6.5*10^14. - Jud McCranie, Dec 02 2019
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			sigma(4) mod (4+3) = 7 mod 7 = 0.

Cf. A000203, A067702, A087167, A088834, A274552, A274553, A274554, A274556.

[n: n in [1..2*10^6] | SumOfDivisors(n) mod (n+3) eq 0 ]; // Vincenzo Librandi, Jul 02 2016

Select[Range[10^6], Mod[DivisorSigma[1, #], # + 3] == 0 &] (* Michael De Vlieger, Jul 01 2016 *)

is(n) = Mod(sigma(n), n+3)==0 \\ Felix Fröhlich, Jul 01 2016

A101223 Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.

Original entry on oeis.org

11, 21, 26, 68, 656, 2336, 8768, 133376, 528896, 34360918016, 35184409837568, 576460757135261696

Offset: 1

Vassil K. Tintschev (tinchev(AT)sunhe.jinr.ru), Dec 15 2004

a(13) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
A subsequence of A274556. a(11) <= b(23) = 35184409837568 ~ 3.5*10^13, since b(k) := 2^(k-1)*(2^k+9) is in this sequence for all k in A057196 (2^k+9 is prime). All known terms except a(2) = 21 are of that form. - M. F. Hasler, Jul 18 2016
Any term x of this sequence can be combined with any term y of A223609 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

			The divisors of 68 are {1, 2, 4, 17, 34, 68} and so sigma(68) = 1 + 2 + 4 + 17+ 24 + 68 = 126 = 2*68 - 10; thus, the deficiency of 68 is 10 so 68 is a term of the sequence.

Cf. A033879, A033880, A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A125248 (deficiency 16).
Cf. also A274556.
Cf. A223609 (abundance 10).

[n: n in [1..9*10^6] | (SumOfDivisors(n)) eq 2*n-10]; // Vincenzo Librandi, Sep 15 2016

Select[ Range[ 85000000], DivisorSigma[1, # ] + 10 == 2# &]

Edited and extended by Robert G. Wilson v, Dec 15 2004
a(10) from Donovan Johnson, Dec 23 2008
Edited by M. F. Hasler, Jul 18 2016
a(11)-a(12) from Hiroaki Yamanouchi, Aug 21 2018

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

A274552 Numbers k such that sigma(k) == 0 (mod k-3).

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 15, 52, 315, 592, 1155, 2102272, 815634435

Offset: 1

Paolo P. Lava, Jun 28 2016

			sigma(4) mod (4-3) = 7 mod 1 = 0.

Contains subsequence A141548.

Cf. A000203, A067702, A087485, A274551, A274553, A274554, A274556, A274557, A274558, A274559, A274560, A274561, A274562, A274563, A274564, A274565, A274566.

[n: n in [1..2*10^6] | n ne 3 and SumOfDivisors(n) mod (n-3) eq 0 ]; // Vincenzo Librandi, Jul 02 2016

k = -3; Select[Range[1, 10^6], # + k != 0 && Mod[DivisorSigma[1, #], # + k] == 0 &] (* Michael De Vlieger, Jul 01 2016 *)

is(n) = if(n == 3, return(0), Mod(sigma(n), n-3)==0) \\ Felix Fröhlich, Jul 02 2016

a(12)-a(13) from Giovanni Resta

a(1)=2 inserted by Max Alekseyev, Jun 08 2025

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

A274558 Numbers k such that sigma(k) == 0 (mod k-6).

Original entry on oeis.org

5, 7, 13, 14, 20, 30, 45, 76, 630, 688, 2310, 8896, 133888, 537051136, 1631268870, 35184418226176, 144115191028645888, 2305843021024854016

Offset: 1

Paolo P. Lava, Jul 05 2016

Contains terms of A141549, odd terms of A141548 multiplied by 2, and 6 times terms of A191363 coprime to 6. - Max Alekseyev, May 25 2025

			sigma(7) mod (7-6) = 8 mod 1 = 0.

Contains subsequence A141549.

Cf. A000203, A045770, A067702, A088833, A141548, A181598, A191363, A274551, A274552, A274553, A274554, A274556, A274557, A274559, A274560, A274561, A274562, A274563, A274564, A274565, A274566.

Select[Range[7, 10^6],  # - 6 != 0 && Mod[DivisorSigma[1, #], # - 6] == 0 &] (* Michael De Vlieger, Jul 05 2016 *)

a(14)-a(15) from Giovanni Resta

Term 5 inserted, a(16)-a(18) added by Max Alekseyev, Jun 04 2025

A274560 Numbers k such that sigma(k) == 0 (mod k-7).

Original entry on oeis.org

3, 5, 6, 8, 10, 11, 15, 27, 34, 72, 232, 34432, 549762629632

Offset: 1

Paolo P. Lava, Jul 05 2016

			sigma(8) mod (8-7) = 15 mod 1 = 0.

Contains subsequence A141550.

Cf. A000203, A045770, A067702, A088833, A181598, A274551, A274552, A274553, A274554, A274556, A274557, A274558, A274559, A274561, A274562, A274563, A274564, A274565, A274566

Select[Range[1, 10^6], Mod[DivisorSigma[1, #], # - 7] == 0 &] (* Michael De Vlieger, Jul 05 2016 *)

a(13) from Giovanni Resta

Terms 3,5,6 inserted by Max Alekseyev, May 29 2025

A274564 Numbers k such that sigma(k) == 0 (mod k-9).

Original entry on oeis.org

6, 7, 8, 10, 11, 15, 19, 24, 33, 105, 33705, 33624064, 2199041081344

Offset: 1

Paolo P. Lava, Jul 06 2016

			sigma(10) mod (10 - 9) = 18 mod 1 = 0.

Contains subsequence A223608.

Cf. A045770, A067702, A088833, A181598, A274551, A274552, A274553, A274554, A274556, A274557, A274558, A274559, A274560, A274561, A274562, A274563, A274565, A274566.

[n: n in [10..2*10^6] | SumOfDivisors(n) mod (n-9) eq 0 ]; // Vincenzo Librandi, Jul 06 2016

k = -9; Select[Range[Abs@k+1, 10^6], Mod[DivisorSigma[1, #], # + k] == 0 &] (* Vincenzo Librandi, Jul 06 2016 *)

isok(k) = (k!=9) && (Mod(sigma(k), k-9) == 0); \\ Michel Marcus, May 30 2025

a(12)-a(13) from Giovanni Resta, Jul 06 2016

Terms 6,7,8 inserted by Max Alekseyev, May 29 2025

A274559 Numbers k such that sigma(k) == 0 (mod k+7).

Original entry on oeis.org

8, 272, 7232, 30848, 516608, 134094848, 2146992128, 35184309174272

Offset: 1

Paolo P. Lava, Jul 05 2016

			sigma(8) mod (8+7) = 15 mod 15 = 0.

Contains subsequence A141546.

Cf. A045770, A067702, A088833, A181598, A274551, A274552, A274553, A274554, A274556, A274557, A274558, A274560, A274561, A274562, A274563, A274564, A274565, A274566.

Select[Range[10^6], Mod[DivisorSigma[1, #], # + 7] == 0 &] (* Michael De Vlieger, Jul 05 2016 *)

a(6)-a(7) from Giovanni Resta, Jul 05 2016

a(8) from Max Alekseyev, May 29 2025

cp's OEIS Frontend

A274551 Numbers k such that sigma(k) == 0 (mod k+3).

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A101223 Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.

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

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A274552 Numbers k such that sigma(k) == 0 (mod k-3).

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

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

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A274558 Numbers k such that sigma(k) == 0 (mod k-6).

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A274560 Numbers k such that sigma(k) == 0 (mod k-7).

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