A141548 -id:A141548 - cp's OEIS frontend

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

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

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

A087485 Odd numbers n such that 2n - sigma(n) = 6.

Original entry on oeis.org

7, 15, 315, 1155, 815634435

Offset: 1

Farideh Firoozbakht, Oct 23 2003

This is a subsequence of A077374.
Except for the first term, all known terms of this sequence are divisible by 15. Is there a number n > 1 such that gcd(a(n),3)=1 or gcd(a(n),5)=1?
a(6) > 10^13. - Giovanni Resta, Mar 29 2013
Also, a subsequence of A141548. - M. F. Hasler, Apr 12 2015
The terms a(3) through a(5) are of the form a(k)*p*q, but I have proved that there is no other term of this form with k <= 5. - M. F. Hasler, Apr 13 2015
The terms are also of the form a(n) = 2*p(n) + 1, with primes p(n) = 3, 7, 157, 577, 407817217. All but the last one are such that 2*p(n) - 1 = a(n) - 2 is again prime. - M. F. Hasler, Nov 27 2016
Terms a(2..5) satisfy 2*a(n) - nextprime(sigma(a(n))) = (-1)^n, see also A067795. - M. F. Hasler, Feb 14 2017

			15 is in the sequence because 2*15-sigma(15)=6.

Cf. A077374, A087167, A088011, A088012.

Do[If[OddQ[n]&&2n-DivisorSigma[1, n]==6, Print[n]], {n, 2*10^9}]

is(n)=bittest(n,0)&&sigma(n)+6==2*n \\ M. F. Hasler, Apr 12 2015

a(3) = a(2)*3*7; a(4) = a(2)*7*11 with 7 = precprime(a(2)*2/3), 11=nextprime(a(2)*2/3); a(5) = a(4)*547*1291. - M. F. Hasler, Apr 13 2015

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

Original entry on oeis.org

1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 127, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137

Offset: 1

Omar E. Pol, Apr 04 2015

Row sums give A002001.
The sum of all terms of first n rows gives A000302(n-1).
The rows of triangle A256263 converge to this sequence.
Rows converge to A016969.
First 11 terms agree with A151548.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
5,7;
5,11,17,15;
5,11,17,23,29,35,41,31;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,131,137,143,149,155,161,167,173,179,185,127;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1    1      |_| | | |_ _  | |_ _ _ _ _ _  | |
2    3      |_ _| | |_  | | |_ _ _ _ _  | | |
3    5      |_ _ _| | | | | |_ _ _ _  | | | |
4    7      |_ _ _ _| | | | |_ _ _  | | | | |
5    5      | | |_ _ _| | | |_ _  | | | | | |
6   11      | |_ _ _ _ _| | |_  | | | | | | |
7   17      |_ _ _ _ _ _ _| | | | | | | | | |
8   15      |_ _ _ _ _ _ _ _| | | | | | | | |
9    5      | | | | | | |_ _ _| | | | | | | |
10  11      | | | | | |_ _ _ _ _| | | | | | |
11  17      | | | | |_ _ _ _ _ _ _| | | | | |
12  23      | | | |_ _ _ _ _ _ _ _ _| | | | |
13  29      | | |_ _ _ _ _ _ _ _ _ _ _| | | |
14  35      | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
15  41      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
16  31      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
It appears that A241717 can be represented by a similar diagram.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A000225, A000302, A002001, A011782, A016969, A141548, A241717, A256260, A256261, A256263, A256264.

Nest[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 7] (* Ivan Neretin, Feb 14 2017 *)

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

A256873 a(n) = 2^(n-1)*(2^n+5).

Original entry on oeis.org

3, 7, 18, 52, 168, 592, 2208, 8512, 33408, 132352, 526848, 2102272, 8398848, 33574912, 134258688, 536952832, 2147647488, 8590262272, 34360393728, 137440264192, 549758435328, 2199028498432, 8796103507968, 35184393060352, 140737530298368, 562950037307392

Offset: 0

M. F. Hasler, Apr 24 2015

For k in A059242, a(k) is in A141548, i.e., A256873 o A059242 is a subsequence of A141548.

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

[2^(n-1)*(2^n+5): n in [0..30]]; // Vincenzo Librandi, Apr 24 2015

Table[2^(n - 1) (2^n + 5), {n, 0, 30}] (* Vincenzo Librandi, Apr 24 2015 *)
LinearRecurrence[{6,-8},{3,7},30] (* Harvey P. Dale, Aug 21 2020 *)

PARI
```
A256873(n)=2^(n-1)*(2^n+5)
```

Vec((3-11*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 26 2015

G.f.: (3-11*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 24 2015

a(n) = 6*a(n-1)-8*a(n-2). - Colin Barker, Apr 26 2015

A326138 Numbers k such that A005187(k) < sigma(k) <= 2k, where A005187(k) = 2k - {binary weight of k}.

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 391612, 15370304, 17619844, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056, 42783299288, 80999455688, 137438691328, 217898810368, 546409576448, 1081071376208, 1661355408388

Offset: 1

Antti Karttunen, Jun 13 2019

nonn

Non-abundant numbers whose deficiency (A033879) is less than their binary weight (A000120).

No other terms below < 2^31.

			815634435 = 3*5*7*11*547*1291 is included as in base-2 (A007088) it is written as 110000100111011001100000000011_2, thus A000120(815634435) = 12, while its nonnegative deficiency (A033879) is 2*815634435 - sigma(815634435) = 6 < 12.

Cf. A000120, A000203, A000396 (subsequence), A005187, A033879, A294898, A295296 (deficiency equals binary weight), A326131, A326132.
Intersection of A263837 and A326133.
Cf. also A087485, A141548, A188597.

isA326138(n) = { my(s=sigma(n)); (s>A005187(n)&&s<=(n+n)); };

a(16)-a(26) from Giovanni Resta, Jun 16 2019

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.

cp's OEIS Frontend

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

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

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A087485 Odd numbers n such that 2n - sigma(n) = 6.

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A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

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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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A256873 a(n) = 2^(n-1)*(2^n+5).

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A326138 Numbers k such that A005187(k) < sigma(k) <= 2k, where A005187(k) = 2k - {binary weight of k}.

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