A101223 -id:A101223 - cp's OEIS frontend

A057196 Numbers k such that 2^k + 9 is prime.

1, 2, 3, 5, 6, 7, 9, 10, 18, 23, 30, 37, 47, 57, 66, 82, 95, 119, 175, 263, 295, 317, 319, 327, 670, 697, 886, 1342, 1717, 1855, 2394, 2710, 3229, 3253, 3749, 4375, 4494, 4557, 5278, 5567, 9327, 10129, 12727, 13615, 14893, 16473, 23639, 40053, 44399, 50335, 80949

Offset: 1

Robert G. Wilson v, Sep 15 2000

nonn

Some of the larger terms are only probable primes.
For these numbers k, 2^(k-1)*(2^k+9) has deficiency 10 (see A101223). - M. F. Hasler, Jul 18 2016
The terms a(48)-a(51) were found by Mike Oakes, a(52) found by Gary Barnes, and a(53-56) found by Lelio R Paula (see link Henri Lifchitz and Renaud Lifchitz). - Elmo R. Oliveira, Dec 01 2023

			For k = 10, 2^10 + 9 = 1033 is prime.
For k = 30, 2^30 + 9 = 1073741833 is prime.

Robert Price, Table of n, a(n) for n = 1..56
Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+9, PRP Top Records.

Cf. A094076, A101223, A104070 (primes of the form 2^k+9). [Klaus Brockhaus, Mar 14 2009]

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), this sequence (2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23). [M. F. Hasler, Jul 18 2016]

Do[ If[ PrimeQ[ 2^n +9 ], Print[n]], { n, 1, 15000 }]

for(n=1, oo, ispseudoprime(2^n+9)&&print1(n", ")) \\ M. F. Hasler, Jul 18 2016

a(48)-a(51) from Mike Oakes, Aug 17 2001

Edited by T. D. Noe, Oct 30 2008

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

Original entry on oeis.org

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

A141548 Numbers n whose deficiency is 6.

Original entry on oeis.org

7, 15, 52, 315, 592, 1155, 2102272, 815634435

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

a(9) > 10^12. - Donovan Johnson, Dec 08 2011
a(9) > 10^13. - Giovanni Resta, Mar 29 2013
a(9) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
For all k in A059242, the number m = 2^(k-1)*(2^k+5) is in this sequence. This yields further terms 2^46*(2^47+5), 2^52*(2^53+5), 2^140*(2^141+5), ... All even terms known so far and the initial 7 = 2^0*(2^1+5) are of this form. All odd terms beyond a(2) are of the form a(n) = a(k)*p*q, k < n. We have proved that there is no further term of this form with the a(k) given so far. - M. F. Hasler, Apr 23 2015
A term n of this sequence multiplied by a prime p not dividing it is abundant if and only if p < sigma(n)/6 = n/3-1. For the even terms 592 and 2102272, there is such a prime near this limit (191 resp. 693571) such that n*p is a primitive weird number, cf. A002975. For a(3)=52, the largest such prime, 11, is already too small. Odd weird numbers do not exist within these limits. - M. F. Hasler, Jul 19 2016
Any term x of this sequence can be combined with any term y of A087167 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) = 7, since 2*7 - sigma(7) = 14 - 8 = 6. - _Timothy L. Tiffin_, Sep 13 2016

Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018.

Cf. A087485 (odd terms).
Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141549 (deficiency 12), A141550 (deficiency 14), A125248 (deficiency 16), A223608 (deficiency 18), A223607 (deficiency 20).
Cf. A087167 (abundance 6).

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

lst={};Do[If[n==Plus@@Divisors[n]-n+6,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == - 6 &] (* Vincenzo Librandi, Sep 14 2016 *)

is(n)=sigma(n)==2*n-6 \\ Charles R Greathouse IV, Apr 23 2015, corrected by M. F. Hasler, Jul 18 2016

a(8) from Donovan Johnson, Dec 08 2011

A125248 Numbers n whose abundance sigma(n)-2n = -16. Numbers n whose deficiency is 16.

Original entry on oeis.org

17, 38, 92, 170, 248, 752, 988, 2528, 8648, 12008, 34688, 63248, 117808, 526688, 531968, 820808, 1292768, 1495688, 2095208, 2112512, 3477608, 4495808, 8419328, 12026888, 13192768, 16102808, 26347688, 29322008, 33653888, 169371008

Offset: 1

Jason G. Wurtzel, Nov 25 2006

When p=2^k+15 is prime (cf. A057197), then 2^(k-1)*p is in this sequence. The terms { 17, 38, 92, 248, 752, 2528, 34688, 531968, 2112512, 8419328, 537116672, 2147975168, ...} are of this from, with k in {1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, ...} = A057197. - M. F. Hasler, Jul 18 2016

Any term x of this sequence can be combined with any term y of A141547 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

			The abundance of 38 = (1+2+19+38)-76 = -16

Donovan Johnson, Giovanni Resta and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..69 (terms <= 10^18, first 43 terms from _Donovan Johnson_ and a(44)-a(51) from _Giovanni Resta_)

Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141549 (deficiency 12), A141550 (deficiency 14), A125248 (this), A223608 (deficiency 18), A223607 (deficiency 20); A141547 (abundance 16).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -16]; // Vincenzo Librandi, Sep 14 2016

Select[Range[1, 10^6], DivisorSigma[1, #] - 2 # == - 16 &] (* Vincenzo Librandi, Sep 14 2016 *)

for(n=1,1000000,if(((sigma(n)-2*n)==-16),print1(n,",")))

a(17) to a(30) from Klaus Brockhaus, Nov 29 2006

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

A223609 Numbers n whose abundance is 10. Sigma(n)-2*n = 10.

Original entry on oeis.org

40, 1696, 518656, 34358296576

Offset: 1

Donovan Johnson, Mar 23 2013

Suggested by N. J. A. Sloane and Robert G. Wilson v.
a(5) > 10^12.
a(5) > 10^13. - Giovanni Resta, Mar 29 2013
a(5) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018
Any term x of this sequence can be combined with any term y of A101223 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

			n = 34358296576. sigma(n)-2*n = 10.

Cf. A000203, A033880, A101223 (deficiency 10).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 10]; // Vincenzo Librandi, Sep 15 2016

Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == 10 &]  (* Vincenzo Librandi, Sep 15 2016 *)

for(n=1, 10^8, if(sigma(n)-2*n==10, print1(n ", ")))

A274556 Numbers k such that sigma(k) == 0 (mod k-5).

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 11, 12, 18, 21, 26, 68, 656, 2336, 8768, 133376, 528896, 34360918016, 35184409837568, 576460757135261696

Offset: 1

Paolo P. Lava, Jun 30 2016

			sigma(6) (mod 6-5) = 12 mod 1 = 0.

Contains subsequence A101223.

Cf. A067702, A223609, A274551, A274552, A274553, A274554, A274557, A274558, A274559, A274560, A274561, A274562, A274563, A274564, A274565, A274566.

[n: n in [1..2*10^6] | n ne 5 and SumOfDivisors(n) mod (n-5) eq 0]; // Vincenzo Librandi, Jul 02 2016

n = -5; Select[Range[1, 10^6], # + n != 0 && Mod[DivisorSigma[1, #], # + n] == 0 &] (* Michael De Vlieger, Jul 01 2016 *)

a(18) from Giovanni Resta, Jul 01 2016

Terms 2,3,4 inserted, a(19)-a(20) added by Max Alekseyev, May 25 2025

A141549 Numbers k whose deficiency is 12: 2k - sigma(k) = 12.

Original entry on oeis.org

13, 45, 76, 688, 8896, 133888, 537051136, 35184418226176, 144115191028645888, 2305843021024854016

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

Numbers n whose abundance is -12. No other terms up to n=100,000,000. - Jason G. Wurtzel, Aug 24 2010
For all k in A102633, the number 2^(k-1)*(2^k+11) is in this sequence. So far all terms except a(2) are of this form. For k = 55, 71, this yields terms 649037107316853651724695645454336, 2787593149816327892704951291908936712585216. - M. F. Hasler, Apr 23 2015; edited by Max Alekseyev, May 27 2025
Any term x = a(m) can be combined with any term y = A141545(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2. Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one. So far, these two sequences have not produced an amicable pair. However, if one is ever found, then it will exhibit x-y = 12. - Timothy L. Tiffin, Sep 13 2016
a(11) > 10^20. - Max Alekseyev, May 27 2025

			a(1) = 13, since 2*13 - sigma(13) = 26 - 14 = 12. - _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), A141550 (deficiency 14), A125248 (deficiency 16), A223608 (deficiency 18), A223607 (deficiency 20); A141545 (abundance 12).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -12]; // Vincenzo Librandi, Sep 14 2016

lst={};Do[If[n==Plus@@Divisors[n]-n+12,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == - 12 &] (* Vincenzo Librandi, Sep 14 2016 *)

for(n=1, 10^8, if(((sigma(n)-2*n)==-12), print1(n, ", "))) \\ Jason G. Wurtzel, Aug 24 2010

a(7) from Donovan Johnson, Dec 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Aug 21 2018
a(10) from Max Alekseyev, May 27 2025

A141550 Numbers n whose deficiency is 14.

Original entry on oeis.org

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

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

cp's OEIS Frontend

A057196 Numbers k such that 2^k + 9 is prime.

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

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A141548 Numbers n whose deficiency is 6.

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A125248 Numbers n whose abundance sigma(n)-2n = -16. Numbers n whose deficiency is 16.

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

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A223609 Numbers n whose abundance is 10. Sigma(n)-2*n = 10.

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A274556 Numbers k such that sigma(k) == 0 (mod k-5).

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