A088833 -id:A088833 - cp's OEIS frontend

A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.

1, 3, 5, 7, 9, 11, 15, 21, 315, 1155, 8925, 32445, 442365, 815634435

Offset: 1

Jason Earls, Nov 30 2002

Apart from {1, 3, 5, 7, 9, 11, 15, 21, 315}, subset of A088012. Probably finite. - Charles R Greathouse IV, Mar 28 2011
a(15) > 10^13. - Giovanni Resta, Mar 29 2013
The abundance of the given terms a(1..14) is: (-1, -2, -4, -6, -5, -10, -6, -10, -6, -6, 6, 6, 6, -6). See also A171929, A188263 and A188597 for numbers with abundancy sigma(n)/n close to 2. - M. F. Hasler, Feb 21 2017
a(15) > 10^22. - Wenjie Fang, Jul 13 2017

			sigma(32445) = 64896 and 32445*2 = 64890, which makes the odd number 32445 six away from perfection: A(32445) = 6 and hence in this sequence.

Eric Weisstein's World of Mathematics, Abundance.

Contains odd terms of: A191363, A088831, A125246, A088832, A141548, A087167, A125247, A088833, A101223, A223609.

Cf. A033879, A033880, A000203, A088012.

Select[Range[1, 10^6, 2], -10 <= DivisorSigma[1, #] - 2 # <= 10 &] (* Michael De Vlieger, Feb 22 2017 *)

forstep(n=1,442365,2,if(abs(sigma(n)-2*n)<=10,print1(n,",")))

a(14) from Farideh Firoozbakht, Jan 12 2004

A125247 Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.

Original entry on oeis.org

22, 130, 184, 1012, 2272, 18904, 33664, 70564, 85936, 100804, 391612, 527872, 1090912, 17619844, 2147713024, 6800695312, 34360655872, 549759483904, 1661355408388, 28502765343364, 82994670582016, 99249696661504, 120646991405056, 431202442356004, 952413274955776

Offset: 1

Jason G. Wurtzel, Nov 25 2006

a(19) > 10^12. - Donovan Johnson, Dec 08 2011
a(20) > 10^13. - Giovanni Resta, Mar 29 2013
a(30) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
a(20) <= 36028797958488064 ~ 3.6*10^16. Indeed, if k is in A057195 then 2^(k-1)*A168415(k) is in this sequence, and k=28 yields this upper bound for a(20) which is in any case a term of this sequence. - M. F. Hasler, Apr 27 2015
If n is in this sequence and p a prime not dividing n, then np is abundant if and only if p < sigma(n)/8 = n/4-1. For all n=a(k) except {22, 70564, 100804, 17619844}, there is such a p near this limit, such that n*p is a primitive weird number (A002975; in A258882 for the terms mentioned in the preceding comment). - M. F. Hasler, Jul 20 2016
Any term x of this sequence can be combined with any term y of A088833 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
Is there any odd number in this sequence? Is it possible to prove the contrary? - M. F. Hasler, Feb 22 2017

			The abundance of 22 = (1+2+11+22)-44 = -8

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..29

Cf. A033880, A088833 (abundance 8).

[n: n in [1..2*10^7] | (DivisorSigma(1,n)-2*n) eq - 8]; // Vincenzo Librandi, Jul 22 2016

Select[Range[10^6], DivisorSigma[1, #] - 2 # == -8 &] (* Michael De Vlieger, Jul 21 2016 *)

for(n=1,1000000,if(((sigma(n)-2*n)==-8),print1(n,",")))

a(13)-a(15) from Klaus Brockhaus, Nov 29 2006
a(16)-a(17) from Donovan Johnson, Dec 23 2008
a(18) from Donovan Johnson, Dec 08 2011
a(19) from Giovanni Resta, Mar 29 2013
a(20)-a(25) from Hiroaki Yamanouchi, Aug 21 2018

A181598 Numbers m with divisor 8 | m and abundance sigma(m)-2*m = 8.

Original entry on oeis.org

56, 368, 11096, 17816, 77744, 128768, 2087936, 2291936, 13174976, 35021696, 45335936, 381236216, 4856970752, 6800228816, 8589344768, 1461083549696, 1471763808896, 2199013818368, 19502341651712, 118123076415296, 933386556194816, 144141575952121856, 417857739454939136

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

a(19) > 10^13. - Giovanni Resta, Apr 02 2014

Cf. A008590, A088833, A097498, A181595, A181596, A181597, A118372, A045768, A000396, A005101, A153501, A005820.

isok(n) = !(n % 8) && (sigma(n) - 2*n == 8); \\ Michel Marcus, Feb 08 2016

A088833 INTERSECT A008590. - R. J. Mathar, Nov 04 2010

Definition rephrased by R. J. Mathar, Nov 04 2010
a(16)-a(17) from Donovan Johnson, Dec 08 2011
a(18) from Giovanni Resta, Apr 02 2014
a(19)-a(23) from the b-file at A088833 added by Amiram Eldar, Mar 11 2024

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

A258885 Primitive weird numbers (A002975) having 6 distinct prime factors.

Original entry on oeis.org

1550860550, 44257207676, 66072609790

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, Jun 14 2015

a(4) <= 5976833582079328 = 2^5*181*197*353*431*34429 and a(5) <= 48083019473926272314825065088 = 2^7*257*97213*97973*100957*1520132521 that is certainly in this sequence. - Giuseppe Melfi, Oct 26 2015

a(4) <= 125258675788784 = 2^4 * 47 * 149 * 353 * 1307 * 2423. - M. F. Hasler, Jul 12 2016

			a(1) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 = A273815(1). (Abundance = 20)
a(2) = 44257207676 = 2^2 * 11 * 37 * 59 * 523 * 881. (Abundance = 8, cf. A088833)
a(3) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 = A273815(3). (Abundance = 4, cf. A088832)

G. Amato, M. F. Hasler, G. Melfi and M. Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, Vol. 7, No. 1 (2016) 153-163.
G. Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, 147 (2015) 508-514; Lemma 2.

Cf. A002975, A258401, A258882, A258883, A258884.

(* copy the terms from A002975, assign them to 'lst' and then *)
Select[ lst, PrimeNu@# == 6 &]

select(w->omega(w)==6, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 12 2016

One more term added and definition corrected by Giuseppe Melfi, Nov 02 2015

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

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

cp's OEIS Frontend

A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.

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A125247 Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.

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A181598 Numbers m with divisor 8 | m and abundance sigma(m)-2*m = 8.

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

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A258885 Primitive weird numbers (A002975) having 6 distinct prime factors.

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

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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).