A125247 -id:A125247 - 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

A257272 a(n) = 2^(n-1)*(2^n+7).

Original entry on oeis.org

4, 9, 22, 60, 184, 624, 2272, 8640, 33664, 132864, 527872, 2104320, 8402944, 33583104, 134275072, 536985600, 2147713024, 8590393344, 34360655872, 137440788480, 549759483904, 2199030595584, 8796107702272, 35184401448960, 140737547075584, 562950070861824, 2251800048566272, 9007199724503040

Offset: 0

M. F. Hasler, Apr 27 2015

For n in A057195, a(n) is of deficiency 8, i.e., in A125247.

Also, the third column (k=2) of the table given in A181444.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000079, A007582, A028403, A256873, A257273, A256871.

Cf. A168415, A057195, A104066, A125247.

[2^(n-1)*(2^n+7): n in [0..25]]; // Vincenzo Librandi, Apr 27 2015

Table[2^(n - 1) (2^n + 7), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(4 - 15 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)

PARI
```
a(n)=2^(n-1)*(2^n+7)
```

Vec((4-15*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

a(n) = 2^(n-1)*A168415(n).
n in A057195 <=> A168415(n) in A104066 <=> a(n) in A125247.
G.f.: (4-15*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

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.

A217769 Least number k > n such that sigma(k) = 2*(k-n), or 0 if no such k exists.

Original entry on oeis.org

6, 3, 5, 7, 22, 11, 13, 27, 17, 19, 46, 23, 124, 58, 29, 31, 250, 57, 37, 55, 41, 43, 94, 47, 1264, 106, 53, 87, 118, 59, 61, 85, 134, 67, 142, 71, 73, 712, 158, 79, 166, 83, 405, 115, 89, 141, 406, 119, 97, 202, 101, 103, 214, 107, 109, 145, 113, 177, 418, 143

Offset: 0

Jayanta Basu, Mar 28 2013

nonn

a(0) = 6 corresponds to the smallest perfect number.
Is n = 144 the first number for which a(n) = 0? - T. D. Noe, Mar 28 2013
No, a(144) = 95501968. - Giovanni Resta, Mar 28 2013
We can instead compute k - sigma(k)/2 for increasing k, which is computationally much faster. In this case, we stop computing when all n have been found for a range of numbers. - T. D. Noe, Mar 28 2013
Also, the first number whose deficiency is 2n. This is the even bisection of A082730. Hence, the first number in the following sequences: A000396, A191363, A125246, A141548, A125247, A101223, A141549, A141550, A125248, A223608, A223607, A223606. - T. D. Noe, Mar 29 2013
10^12 < a(654) <= 618970019665683124609613824. - Donovan Johnson, Jan 04 2014

			a(4)=22, since 22 is the least number such that sigma(22)=36=2*(22-4).

Donovan Johnson, Table of n, a(n) for n = 0..653 (first 241 terms from T. D. Noe)
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.

Cf. A000203, A000396.

Cf. A087998 (negative n).

Table[Min[Select[Range[2000], DivisorSigma[1, #] == 2*(# - i) &]], {i, 0, 60}]
nn = 144; t = Table[0, {nn}]; k = 0; While[k++; Times @@ t == 0, s = (2*k - DivisorSigma[1, k])/2; If[s >= 0 && s < nn && IntegerQ[s] && t[[s + 1]] == 0, t[[s + 1]] = k]]; t (* T. D. Noe, Mar 28 2013 *)

A088820 Numbers k with abundance radius of 8, i.e., abs(sigma(k)-2*k) = 8.

Original entry on oeis.org

22, 56, 130, 184, 368, 836, 1012, 2272, 11096, 17816, 18904, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 128768, 254012, 388076, 391612, 527872, 1090912, 2087936, 2291936, 13174976, 17619844, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216

Offset: 1

Labos Elemer, Oct 20 2003

nonn

Original definition: Abundance-radius=8, that is Abs[sigma[n]-2n]=8 (either +8 or -8). A045770 from 3rd term complemented by -8 cases.

			22 is in the sequence since sigma(22) = 1 + 2 + 11 + 22 = 36 = 2*22 - 8.
56 is in the sequence since sigma(56) = 1 + 2 + 4 + 7 + 8 + 14 + 28 + 56 = 120 = 2*56 + 8. - _Michael B. Porter_, Jul 20 2016

Amiram Eldar, Table of n, a(n) for n = 1..66 (terms below 10^18, from the b-files at A088833 and A125247)

Disjoint union of A088833 (abundance 8) and A125247 (deficiency 8).
Cf. A045770, A045768, A055708, A077374, A088007, A088008, A088010, A088011, A088012, A088818, A088819.
Cf. A000203 (sigma), A033880 (abundance), A005100 (deficient numbers).

[n: n in [1..2*10^7] | Abs(DivisorSigma(1, n) - 2*n) eq 8]; // Vincenzo Librandi, Jul 20 2016

Select[Range[1, 10^6], Abs[DivisorSigma[1, #] - 2 #] == 8 &] (* Vincenzo Librandi, Jul 20 2016 *)

is(n)=abs(sigma(n)-2*n)==8 \\ Use, e.g., select(is,[1..10^5]*2). - M. F. Hasler, Jul 19 2016

More terms from David Wasserman, Aug 18 2005
Edited by M. F. Hasler, Jul 19 2016
a(33)-a(34) from Amiram Eldar, Mar 11 2025

A292557 a(n) is the smallest number k such that 2k - sigma(k) = 2^n.

Original entry on oeis.org

3, 5, 22, 17, 250, 134, 262, 257, 6556, 4124, 10330, 8198, 91036, 19649, 65542, 65537, 1442716, 524294, 1363258, 4194332, 4411642, 16442342, 16866106, 22075325, 156791188, 536871032, 2160104368, 536870918, 1074187546, 2147483654, 4295862586, 19492545788

Offset: 1

XU Pingya, Sep 19 2017

nonn

Primes of the form 2^n+1, i.e., Fermat primes (A019434) are terms of this sequence.

For n > 32, a(n) > 2 * 10^10.

			sigma(20) - 2*20 = 2^1, a(1) = 20.
sigma(108) - 2*108 = 64 = 2^6, a(6) = 108.

Cf. A019434, A033879, A125246, A125247, A125248, A191363, A275997.

Table[k = 1; While[Log[2, 2k - DivisorSigma[1, k]] != n, k++]; k, {n, 31}]

a(n) = my(k=1); while(2*k - sigma(k) != 2^n, k++); k; \\ Michel Marcus, Sep 19 2017

A162302 Numbers n such that (A000203(n)+28)/n is an integer.

Original entry on oeis.org

1, 28, 29, 62, 84, 182, 230, 252, 344, 756, 944, 2268, 6710, 6804, 20264, 20412, 36224, 61236, 183708, 538112, 551124, 1653372, 2085710, 4960116, 14503550, 14880348, 33665024, 44641044, 55328384, 133923132, 134438912, 401769396, 615206030, 1082574464

Offset: 1

Ctibor O. Zizka, Jun 30 2009

nonn

Contains the subset of all n of the form 28*3^k.
Generalized sequences are defined by A*A000203(n)+ B = C*n with A,B,C integers.
Then we get for different settings of A, B, C hyperperfect numbers:
A=1, C=2, B=0 gives A000396. A=1, C=2, B=1 gives A000079.
A=1, C=2, B=2 gives A056006. A=1, C=2, B=4 gives A125246. A=1, C=2, B=6 gives A141548.
A=1, C=2, B=8 gives A125247. A=1, C=2, B=10 gives A101223. A=1, C=2, B=12 gives A141549.
A=1, C=2, B=14 gives A141550. A=1, C=2, B=16 gives A125248. A=1, C=2, B=0 gives A000396.
A=1, C=2, B=0 gives A000396. A=1, C=3, B=0 gives A005820.
Not in the OEIS: A=1, C=3, B=12,18,28,... A=2, C=3, B=21,27,33,45,... A=3, C=4, B=20,...
Terms not of the form 28*3^n: 1, 29, 62, 182, 230, 344, 944, 6710, 20264, 36224, 538112, 2085710, 14503550, 33665024, 55328384, ..., . [Robert G. Wilson v, Sep 05 2010]

Graeme L. Cohen, Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol.5 (1996), No. 2, pp.91-100.

Cf. A000203, A000396, A000079, A005820, A056006, A125246, A141548, A125247, A101223, A141549, A141550, A125248,

A000203 := proc(n) numtheory[sigma](n) ; end proc:
isA152302 := proc(n) (A000203(n)+28) mod n = 0 ; end proc:
for n from 1 to 1000000 do if isA152302(n) then printf("%d,",n) ; end if ; end do: # R. J. Mathar, Aug 25 2010

fQ[n_] := Divisible[ DivisorSigma[1, n] + 28, n]; lst = {}; k = 1; While[k < 10^9/4, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Sep 05 2010 *)

Missing terms (1, 29, 182,..) inserted, 7 terms added, comment corrected - R. J. Mathar, Aug 25 2010
a(22)-a(30) from Robert G. Wilson v, Sep 05 2010
a(31)-a(34) from Donovan Johnson, Nov 03 2011

cp's OEIS Frontend

A141550 Numbers n whose deficiency is 14.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

Extensions

A257272 a(n) = 2^(n-1)*(2^n+7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

A217769 Least number k > n such that sigma(k) = 2*(k-n), or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A088820 Numbers k with abundance radius of 8, i.e., abs(sigma(k)-2*k) = 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma