A045770 -id:A045770 - cp's OEIS frontend

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

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

A274561 Numbers k such that sigma(k) == 0 (mod k+8).

Original entry on oeis.org

10, 49, 240, 550, 748, 1504, 3192, 7192, 7912, 10792, 17272, 30592, 979992, 1713592, 4526272, 8353792, 9928792, 11547352, 17999992, 89283592, 173482552, 361702144, 1081850752, 1845991216, 2146926592, 11097907192, 12985220152, 21818579968, 34357510144, 109170719992, 228354264064, 279632332792, 549746900992, 1511712719992, 2169800814592

Offset: 1

Paolo P. Lava, Jul 05 2016

nonn

			sigma(10) mod (10 + 8) = 18 mod 18 = 0.

Cf. A000203, A045770, A067702, A088833, A181598, A274551-A274560, A274562-A274566.

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

a(16)-a(35) from Giovanni Resta

A274563 Numbers k such that sigma(k) == 0 (mod k+9).

Original entry on oeis.org

15, 208, 6976, 8415, 31815, 351351, 2077696, 20487159, 159030135, 536559616, 2586415095, 137433972736, 2199003332608

Offset: 1

Paolo P. Lava, Jul 06 2016

			sigma(15) mod (15 + 9) = 24 mod 24 = 0.

Cf. A045770, A067702, A088833, A181598, A223610, A274551-A274562, A274564-A274566.

Select[Range[10^6], Mod[DivisorSigma[1, #], # + 9] == 0 &] (* Michael De Vlieger, Jul 06 2016 *)

a(7)-a(13) from Giovanni Resta, Jul 06 2016

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

A088834 Numbers k such that sigma(k) == 6 (mod k).

Original entry on oeis.org

1, 5, 6, 25, 180, 8925, 32445, 442365

Offset: 1

Labos Elemer, Oct 29 2003

For each integer j in A059609, 2^(j-1)*(2^j - 7) is in the sequence. E.g., for j = A059609(1) = 39 we get 151115727449904501489664. - M. F. Hasler and Farideh Firoozbakht, Dec 03 2013
No more terms to 10^10. - Charles R Greathouse IV, Dec 05 2013
a(9) > 10^13. - Giovanni Resta, Apr 02 2014
a(9) > 1.5*10^14. - Jud McCranie, Jun 02 2019
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			Sigma(25) = 31 = 1*25 + 6, so 31 mod 25 = 6.

Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. Article ID 10.3.1.

Cf. A054024, A045768, A045769, A045770, A077374, A076496, A088012.
Cf. A087167 (a subsequence).
Cf. A059609.

Select[Range[1000000], Mod[DivisorSigma[1, #] - 6, #] == 0 &] (* T. D. Noe, Dec 03 2013 *)

isok(n) = Mod(sigma(n), n) == 6; \\ Michel Marcus, Jan 03 2023

Terms corrected by Charles R Greathouse IV and Farideh Firoozbakht, Dec 03 2013

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

A274565 Numbers k such that sigma(k) == 0 (mod k+10).

Original entry on oeis.org

14, 176, 1376, 3230, 3770, 6848, 114256, 125696, 544310, 561824, 740870, 2075648, 4199030, 4607296, 8436950, 33468416, 134045696, 199272950, 624032630, 1113445430, 1550860550, 85905593344, 2199001235456, 35184284008448

Offset: 1

Paolo P. Lava, Jul 06 2016

			sigma(14) mod (14 + 10) = 24 mod 24 = 0.

Contains subsequence A223611.
Contains 2 * odd terms of A223609.
Cf. A045770, A067702, A088833, A181598, A274551, A274552, A274553, A274554, A274556, A274557, A274558, A274559, A274560, A274561, A274562, A274563, A274564, A274566.

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

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

a(13)-a(23) from Giovanni Resta, Jul 06 2016

a(24) from Max Alekseyev, May 29 2025

A117346 Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 16, 17, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 70, 71, 73, 79, 83, 88, 89, 97, 101, 103, 104, 107, 109, 110, 113, 120, 127, 128, 131, 136, 137, 139, 149, 151, 152, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Sigma is the sum_of_divisors function.

			70 is in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiperfect Number.

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117347 through A117350.

asmlQ[n_]:=Module[{p=Mod[DivisorSigma[1,n],n]},If[p>n/2,p=n-p];p<=Log[n]];
Select[Range[200],asmlQ] (* Harvey P. Dale, Dec 25 2013 *)

First term prepended by Harvey P. Dale, Dec 25 2013

A117349 Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

Original entry on oeis.org

6, 10, 20, 28, 70, 88, 104, 110, 120, 136, 152, 464, 496, 592, 650, 672, 884, 1155, 1888, 1952, 2144, 4030, 5830, 8128, 8384, 8925, 11096, 17816, 18632, 18904, 30240, 32128, 32445, 32760, 32896, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 116624

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Log is the natural logarithm. Sigma is the sum_of_divisors function.

			70 is a term because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

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

Donovan Johnson, Table of n, a(n) for n = 1..180 (terms <= 10^12)
Eric Weisstein's World of Mathematics, Multiperfect Number

Cf. A045768, A045769, A045770, A077374, A087167, A087485.
Cf. A088007, A088008, A088009, A088010, A088011, A088012.
Cf. A117346, A117347, A117348, A117350.

sigma(n) = k*n + r, abs(r) <= log(n).

Offset corrected by Donovan Johnson, Oct 01 2012

cp's OEIS Frontend

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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A274561 Numbers k such that sigma(k) == 0 (mod k+8).

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A274563 Numbers k such that sigma(k) == 0 (mod k+9).

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

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A088834 Numbers k such that sigma(k) == 6 (mod k).

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

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A274565 Numbers k such that sigma(k) == 0 (mod k+10).

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