A141546 -id:A141546 - cp's OEIS frontend

A141550 Numbers n whose deficiency is 14.

27, 34, 232, 34432, 549762629632

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

nonn

a(6) > 10^12. - Donovan Johnson, Dec 08 2011
a(6) > 10^13. - Giovanni Resta, Mar 29 2013
a(6) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
a(6) <= b(38) = 37778931864743868104704 = 3.77789*10^22, since b(k) = 2^(k-1)*(2^k+13) is in this sequence for all k in A102634, i.e., 2^k+13 is prime. All known terms except a(1) = 27 are of this form: a(2..5) = b(k) with k = 2, 4, 8, 20, and k = 38 yields the next larger term of this form. - M. F. Hasler, Jul 18 2016
Any term x of this sequence can be combined with any term y of A141546 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

			a(1) = 27, since 2*27 - sigma(27) = 54 - 40 = 14. - _Timothy L. Tiffin_, Sep 13 2016

Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141549 (deficiency 12), A125248 (deficiency 16); A141546 (abundance 14).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -14]; // Vincenzo Librandi, Sep 14 2016

lst={};Do[If[n==Plus@@Divisors[n]-n+14,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == - 14 &] (* Vincenzo Librandi, Sep 14 2016 *)

a(5) from Donovan Johnson, Dec 08 2011

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

A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

Original entry on oeis.org

134, 284, 410, 632, 1292, 1628, 4064, 9752, 12224, 22712, 66992, 72944, 403988, 556544, 2161664, 2330528, 8517632, 13228352, 14563832, 15422912, 20732792, 89472632, 134733824, 150511232, 283551872, 537903104, 731670272, 915473696, 1846850576, 2149548032, 2159587616

Offset: 1

Timothy L. Tiffin, Aug 16 2016

Any term x = a(m) in this sequence can be used with any term y in A275996 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.
The smallest amicable pair is (220, 284) = (A275996(2), a(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (A275996(7), a(11)) = (A063990(18), A063990(19)), where 66992 - 66928 = 64 is the deficiency of 66992 and the abundance of 66928.
Contains numbers 2^(k-1)*(2^k + 63) whenever 2^k + 63 is prime. - Max Alekseyev, Aug 27 2025

			a(1) = 134, since 2*134 - sigma(134) = 268 - 204 = 64.

Max Alekseyev, Table of n, a(n) for n = 1..89

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A002025, A063990.

Select[Range[10^7], 2 # - DivisorSigma[1, #] == 64 &] (* Michael De Vlieger, Jan 10 2017 *)

isok(n) = 2*n - sigma(n) == 64; \\ Michel Marcus, Dec 30 2016

a(23)-a(31) from Jinyuan Wang, Mar 02 2020

A292626 Numbers k whose abundance is 128: sigma(k) - 2*k = 128.

Original entry on oeis.org

860, 5336, 6536, 9656, 16256, 55796, 70864, 98048, 361556, 776096, 2227616, 4145216, 4498136, 4632896, 8124416, 13086016, 34869056, 38546576, 150094976, 172960856, 196066256, 962085536, 1080008576, 1733780336, 1844788112, 2143256576, 2531343872, 2986104064, 9677743616, 11276687456, 17104503968, 20680182272, 21568135616

Offset: 1

Fabian Schneider, Sep 20 2017

Max Alekseyev, Table of n, a(n) for n = 1..87

Subsequence of A259174.
Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64).

fQ[n_] := DivisorSigma[1, n] == 2 n + 128; Select[ Range@ 10^8, fQ] (* Robert G. Wilson v, Nov 19 2017 *)

isok(n) = sigma(n) - 2*n == 128; \\ Michel Marcus, Sep 20 2017

a(9)-a(18) from Michel Marcus, Sep 20 2017
a(19)-a(24), a(26), a(29)-a(30), a(33) from Robert G. Wilson v, Nov 20 2017
Missing terms a(25), a(27)-a(28), a(31)-a(32) inserted and terms a(34) onward added by Max Alekseyev, Aug 30 2025

A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

Original entry on oeis.org

124, 9664, 151115727458150838697984

Offset: 1

Max Alekseyev, Jul 29 2025

Contains numbers 2^(k-1)*(2^k + 23) for k in A057203. First three terms have this form.

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A275702 (k=26).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26).
Cf. A057203.

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

Original entry on oeis.org

250, 376, 1276, 12616, 20536, 396916, 801376, 1297312, 8452096, 33721216, 40575616, 59376256, 89397016, 99523456, 101556016, 150441856, 173706136, 269096704, 283417216, 500101936, 1082640256, 1846506832, 15531546112, 34675557856, 136310177392, 136783784608

Offset: 1

Max Alekseyev, Aug 27 2025

Contains numbers 2^(k-1)*(2^k + 31) for k in A247952.

Max Alekseyev, Table of n, a(n) for n = 1..54

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A247952.

A101260 Numbers n whose abundance is 56.

Original entry on oeis.org

84, 140, 224, 308, 364, 476, 532, 644, 812, 868, 1036, 1148, 1204, 1316, 1372, 1484, 1652, 1708, 1876, 1988, 2044, 2212, 2324, 2492, 2716, 2828, 2884, 2996, 3052, 3164, 3556, 3668, 3836, 3892, 4172, 4228, 4396, 4544, 4564, 4676, 4844, 5012, 5068, 5348

Offset: 1

Vassil K. Tintschev (tinchev(AT)sunhe.jinr.ru), Dec 17 2004

If n is of the form p*28, where p is a prime distinct from 2 or 7 then n is in this sequence, note that 28 is a perfect number. The terms in the sequence but not divisible by 28 are 4544, 9272, 14552, 25472, 74992, 495104... - Enrique Pérez Herrero, Apr 15 2012

If p=2^k-57 is prime (cf. A165778), then 2^(k-1)*p is in the sequence: For the first such k=6,7,8,10,16,19,22,28,..., this yields 224, 4544, 25472, 495104, 2145615872, 137424011264, 8795973484544, 36028789368553472, ... - M. F. Hasler, Apr 15 2012

			84 is a term of the sequence because 2*2*3*7 = 84 and 84 - 42 - 28 - 21 - 14 - 12 - 7 - 6 - 4 - 3 - 2 = g(84) = -55.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..3000
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A005101, A033880.

Cf. A141545, A141546.

[n: n in [1..10^4] |DivisorSigma(1,n) eq 2*n+56]; // Vincenzo Librandi, Jul 30 2015

Select[ Range[5500], DivisorSigma[1, # ] == 2# + 56 &] (* Robert G. Wilson v, Dec 22 2004 *)

A063788 Numbers k such that sigma(k) = 2k + Omega(k), where Omega(n) is the number of prime divisors of n (with repetition).

Original entry on oeis.org

18, 88, 4030, 5830, 518656, 13174976, 134094848, 2146926592, 2251798907715584, 12504224434300196, 324257317741920256

Offset: 1

Jason Earls, Aug 16 2001

Includes terms 633825300114085990300727115776 and 2596148429267411760623818083663872. - Donovan Johnson, Dec 19 2008; edited by Max Alekseyev, May 27 2025

Terms a(2)-a(4) come from A088832, a(5) from A223609, a(6) and a(10) from A088833, a(7) from A141546, a(8) from A141547, a(9) from A275701, a(11) from A223611. Also includes the following terms k with Omega(k) = 56: 246434407522188377975875310632234056969345758857269346304, 15937923506379504700185810932457673797717574263217988829184, 264936582814027097239593278653623212574863771975442952634761216, 7948097484419456643668355219907727481405487440330234556835692544. - Max Alekseyev, May 27 2025

Cf. A000203 (sigma), A001222 (Omega), A088832, A223609, A088833, A141546, A141547, A275701, A223611.

for(n=1,10^8, if(sigma(n)==2*n+bigomega(n),print(n)))

Numbers k such that A000203(k) = 2k + A001222(k). - Wesley Ivan Hurt, Oct 30 2022

a(7)-a(8) from Donovan Johnson, Dec 19 2008

a(9) from Donovan Johnson confirmed and a(10)-a(11) added by Max Alekseyev, May 27 2025

cp's OEIS Frontend

A141550 Numbers n whose deficiency is 14.

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

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A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

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A292626 Numbers k whose abundance is 128: sigma(k) - 2*k = 128.

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A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

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A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

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A101260 Numbers n whose abundance is 56.

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A063788 Numbers k such that sigma(k) = 2k + Omega(k), where Omega(n) is the number of prime divisors of n (with repetition).

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