A141549 -id:A141549 - cp's OEIS frontend

A102633 Numbers k such that 2^k + 11 is prime.

1, 3, 5, 7, 9, 15, 23, 29, 31, 55, 71, 77, 297, 573, 1301, 1555, 1661, 4937, 5579, 6191, 6847, 6959, 19985, 26285, 47093, 74167, 149039, 175137, 210545, 240295, 306153, 326585, 345547

Offset: 1

Lei Zhou, Jan 20 2005

a(34) > 5*10^5. - Robert Price, Aug 26 2015

For numbers k in this sequence, 2^(k-1)*(2^k+11) has deficiency 12 (see A141549). All terms are odd since 4^n+11 == 1+2 == 0 (mod 3). - M. F. Hasler, Jul 18 2016

			k = 1: 2^1 + 11 = 13 is prime.
k = 3: 2^3 + 11 = 19 is prime.
k = 2: 2^2 + 11 = 15 is not prime.

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+11, PRP Top Records.
Lei Zhou, Between 2^n and primes. [broken link]

Cf. A094076, A141549.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), this sequence (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).

Do[ If[ PrimeQ[2^n + 11], Print[n]], {n, 15250}] (* Robert G. Wilson v, Jan 21 2005 *)

for(n=1,9e9,ispseudoprime(2^n+11)&&print1(n",")) \\ M. F. Hasler, Jul 18 2016

a(18)-a(22) from Robert G. Wilson v, Jan 21 2005
a(23)-a(33) from Robert Price, Dec 06 2013
Edited by M. F. Hasler, Jul 18 2016

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

A141545 Numbers k whose abundance is 12: sigma(k) - 2*k = 12.

Original entry on oeis.org

24, 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

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

nonn

Numbers k such that sigma(k) = 2k + 12. - Wesley Ivan Hurt, Jul 11 2013
Any term x = a(m) can be combined with any term y = A141549(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 y-x = 12. - Timothy L. Tiffin, Sep 13 2016
From Tomohiro Yamada, Jan 01 2023: (Start)
6p belongs to this sequence if p > 3 is prime since sigma(6p) = 12(p + 1) = 12p + 12.  Moreover, 2^m * (2^(m+1) - 13) is also a term of this sequence if 2^(m+1) - 13 is prime (m+1 is a term of A096818) since sigma(2^m * (2^(m+1) - 13)) = (2^(m+1) + 1) * (2^(m+1) - 13) = 2^(m+1) * (2^(m+1) - 13) + 12.  So 24, 304, 127744, 33501184, and 8589082624 also belong to this sequence.
Problem: is 54 the only term of this sequence which is of neither type given above? (End)

			30 is in the sequence since sigma(30) = sigma(2*3*5) = sigma(2)*sigma(3)*sigma(5) = 3*4*6 = 72 = 2(30)+12.  Since this is the second such number whose abundance is 12, a(2) = 30. - _Wesley Ivan Hurt_, Jul 11 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A000203, A005101, A141549 (deficiency 12).

Cf. A076496 (sigma(k) - a*k = 12).

[n: n in [1..1400] | (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, 10^4], DivisorSigma[1, #] - 2 # == 12 &] (* Vincenzo Librandi, Sep 14 2016 *)

is(n)=sigma(n)==2*n+12 \\ Charles R Greathouse IV, Feb 21 2017

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

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

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

A256871 a(n) = 2^(n-1)*(2^n+11).

Original entry on oeis.org

6, 13, 30, 76, 216, 688, 2400, 8896, 34176, 133888, 529920, 2108416, 8411136, 33599488, 134307840, 537051136, 2147844096, 8590655488, 34361180160, 137441837056, 549761581056, 2199034789888, 8796116090880, 35184418226176, 140737580630016, 562950137970688

Offset: 0

M. F. Hasler, Apr 24 2015

a(A102633(n)) is a subsequence of A141549.

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+11): n in [0..30]]; // Vincenzo Librandi, Apr 24 2015

Table[2^(n - 1) (2^n + 11), {n, 0, 30}] (* Vincenzo Librandi, Apr 24 2015 *)
LinearRecurrence[{6,-8},{6,13},40] (* Harvey P. Dale, Jan 29 2022 *)

PARI
```
A256871(n)=2^(n-1)*(2^n+11)
```

Vec((6-23*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 26 2015

G.f.: (6-23*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

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.

cp's OEIS Frontend

A102633 Numbers k such that 2^k + 11 is prime.

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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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A141545 Numbers k whose abundance is 12: sigma(k) - 2*k = 12.

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A141550 Numbers n whose deficiency is 14.

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

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

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