A141548 -id:A141548 - cp's OEIS frontend

A059242 Numbers k such that 2^k + 5 is prime.

1, 3, 5, 11, 47, 53, 141, 143, 191, 273, 341, 16541, 34001, 34763, 42167, 193965, 282203

Offset: 1

Tony Davie (ad(AT)dcs.st-and.ac.uk), Jan 21 2001

The subsequence of primes starts 3, 5, 11, 47, 53, 191, ... - Vincenzo Librandi, Aug 07 2010
For k in this sequence, 2^(k-1)*(2^k+5) is in A141548: numbers of deficiency 6. - M. F. Hasler, Apr 23 2015
a(18) > 5*10^5. - Robert Price, Aug 23 2015
a(18) > 6*10^5. - Tyler NeSmith, Jan 18 2021
All terms are odd - Elmo R. Oliveira, Dec 01 2023

			2^3 + 5 = 13 is prime, but 2^4 + 5 = 21 is not.

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+5, PRP Top Records.

Cf. A094076, A141548.

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

Select[Range[20000],PrimeQ[2^#+5]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)

is(n)=ispseudoprime(2^n+5) \\ M. F. Hasler, Apr 23 2015

More terms from Santi Spadaro, Oct 04 2002
a(12) from Hans Havermann, Oct 07 2002
a(13)-a(15) from Charles R Greathouse IV, Oct 07 2011
a(16)-a(17) from Robert Price, Dec 06 2013

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

A191363 Numbers m such that sigma(m) = 2*m - 2.

Original entry on oeis.org

3, 10, 136, 32896, 2147516416

Offset: 1

Luis H. Gallardo, May 31 2011

Let k be a nonnegative integer such that F(k) = 2^(2^k) + 1 is prime (a Fermat prime A019434), then m = (F(k)-1)*F(k)/2 appears in the sequence.
Conjecture: a(1)=3 is the only odd term of the sequence.
Conjecture: All terms of the sequence are of the above form derived from Fermat primes.
The sequence has 5 (known) terms in common with sequences A055708 (k-1 | sigma(k)) and A056006 (k | sigma(k)+2) since {a(n)} is a subsequence of both.
The first five terms of the sequence are respectively congruent to 3, 4, 4, 4, 4 modulo 6.
After a(5) there are no further terms < 8*10^9.
Up to m = 1312*10^8 there are no further terms in the class congruent to 4 modulo 6.
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
See A125246 for numbers with deficiency 4, i.e., sigma(m) = 2*m - 4, and A141548 for numbers with deficiency 6. - M. F. Hasler, Jun 29 2016 and Jul 17 2016
A term m of this sequence multiplied by a prime p not dividing it is abundant if and only if p < m-1. For each of a(2..5) there is such a prime near this limit (here: 7, 127, 30197, 2147483647) such that a(k)*p is a primitive weird number, cf. A002975. - M. F. Hasler, Jul 19 2016
Any term m of this sequence can be combined with any term j of A088831 to satisfy the property (sigma(m) + sigma(j))/(m+j) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. [Proof: If m = a(n) and j = A088831(k), then sigma(m) = 2m-2 and sigma(j) = 2j+2. Thus, sigma(m) + sigma(j) = (2m-2) + (2j+2) = 2m + 2j = 2(m+j), which implies that (sigma(m) + sigma(j))/(m+j) = 2(m+j)/(m+j) = 2.] - Timothy L. Tiffin, Sep 13 2016
At least the first five terms are a subsequence of A295296 and of A295298. - David A. Corneth, Antti Karttunen, Nov 26 2017
Conjectures: all terms are second hexagonal numbers (A014105). There are no terms with middle divisors. - Omar E. Pol, Oct 31 2018
The symmetric representation of sigma(m) of each of the 5 numbers in the sequence consists of 2 parts of width 1 that meet at the diagonal (subsequence of A246955). - Hartmut F. W. Hoft, Mar 04 2022
The first five terms coincide with the sum of two successive terms of A058891. The same is not true for a(6), if such exists. - Omar E. Pol, Mar 03 2023

			For n=1, a(1) = 3 since sigma(3) = 4 = 2*3 - 2.

Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, and 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. A000203, A002975, A056006, A055708, A088831 (abundance 2).
Cf. A033880, A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A125248 (deficiency 16).
Cf. A295296, A295298.
Cf. A014105, A237593, A246955.
Cf. A058891.

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

ok[n_] := DivisorSigma[1,n] == 2*n-2; Select[ Table[ 2^(2^k-1) * (2^(2^k)+1), {k, 0, 5}], ok] (* Jean-François Alcover, Sep 14 2011, after conjecture *)
Select[Range[10^6], DivisorSigma[1, #] == 2 # - 2 &] (* Michael De Vlieger, Sep 14 2016 *)

zp(a,b) = {my(c,c1,s); c = a; c1 = 2*c-2;
while(c

a(k)=(2^2^k+1)<<(2^k-1) \\ For k<6. - M. F. Hasler, Jul 27 2016

a(n) = (A019434(n)-1)*A019434(n)/2 for all terms known so far. - M. F. Hasler, Jun 29 2016

A125246 Numbers m whose abundance sigma(m) - 2m = -4. Numbers whose deficiency is 4.

Original entry on oeis.org

5, 14, 44, 110, 152, 884, 2144, 8384, 18632, 116624, 8394752, 15370304, 73995392, 536920064, 2147581952, 34360131584, 27034175140420610, 36028797421617152, 576460753914036224

Offset: 1

Jason G. Wurtzel, Nov 25 2006

a(17) > 10^12. - Donovan Johnson, Dec 08 2011
a(17) > 10^13. - Giovanni Resta, Mar 29 2013
a(17) <= b(28) = 36028797421617152 ~ 3.6*10^16, since b(k) := 2^(k-1)*(2^k+3) is in this sequence for all k in A057732, i.e., whenever 2^k+3 is prime, and 28 = A057732(11). Further terms of this form are b(30), b(55), b(67), b(84), ... The only terms not of the form b(k), below 10^13, are {110, 884, 18632, 116624, 15370304, 73995392}. - M. F. Hasler, Apr 27 2015, edited on Jul 17 2016
See A191363 for numbers with deficiency 2, and A141548 for numbers with deficiency 6. - M. F. Hasler, Jun 29 2016 and Jul 17 2016
A term of this sequence multiplied with a prime p not dividing it is abundant if and only if p < sigma(a(n))/4. For each of a(2..16) there is such a prime, near this limit, such that a(n)*p is a primitive weird number, cf. A002975. - M. F. Hasler, Jul 17 2016
Any term x of this sequence can be combined with any term y of A088832 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
Is 5 the only odd number in this sequence? Is it possible to prove this? - M. F. Hasler, Feb 22 2017
a(20) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
If m is an even term, then (m-2)/2 is a term of A067680. - Jinyuan Wang, Apr 08 2020

			The abundance of 5 = (1+5)-10 = -4.
More generally, whenever p = 2^k + 3 is prime (as p = 5 for k = 1), then A(2^(k-1)*p) = (2^k-1)*(p+1) - 2^k*p = 2^k - p - 1 = -4.

Cf. A033880, A057732, A067680, A191363, A141548, A125247, A125248, A088832 (abundance 4).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -4]; // Vincenzo Librandi, Sep 15 2016

Select[Range[10^7], DivisorSigma[1, #] - 2 # == -4 &] (* Michael De Vlieger, Jul 18 2016 *)

for(n=1,1000000,if(((sigma(n)-2*n)==-4),print1(n,",")))

a(11) to a(14) from Klaus Brockhaus, Nov 29 2006
a(15)-a(16) from Donovan Johnson, Dec 23 2008
a(17)-a(19) from Hiroaki Yamanouchi, Aug 21 2018

A087167 Odd numbers such that sigma(n) - 2n = 6.

Original entry on oeis.org

8925, 32445, 442365

Offset: 1

Farideh Firoozbakht, Oct 19 2003

If m is in this sequence and 5 doesn't divide m then m is an odd Weird number. There are no other terms up to 2*10^9. Jud McCranie wrote: There are no terms between 2*10^9 and 6.5*10^9.
a(4) > 10^12. - Donovan Johnson, Dec 08 2011
a(4) > 10^13. - Giovanni Resta, Mar 29 2013
a(4) > 10^22. - Wenjie Fang, Jun 16 2014
Any term x of this sequence can be combined with any term y of A141548 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)=8925 because sigma(8925)=2*8925+6 and 8925 is the first odd number such that sigma(n)-2n=6.

R. K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." B2 in Unsolved Problems in Number Theory, 2nd ed.New York:Springer- Verlag, pp. 45-53, 1994.

S. Benkoski, Are All Weird Numbers Even?, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774.
C. Rivera, Puzzle 233. A little twist.
Eric Weisstein's World of Mathematics, Weird Number.

Cf. A003380, A077374, A005101, A005835, A141548 (deficiency 6).

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

is(n)=n%2 && sigma(n)==2*n+6 \\ Charles R Greathouse IV, Mar 09 2014

A295296 Numbers n such that the sum of their divisors + the number of ones in their binary expansion = 2n; numbers for which A000203(n) + A000120(n) = 2n.

Original entry on oeis.org

1, 2, 3, 4, 8, 10, 16, 32, 64, 128, 136, 256, 315, 512, 1024, 2048, 4096, 8192, 16384, 32768, 32896, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 2147516416

Offset: 1

Antti Karttunen, Nov 26 2017

Numbers n such that their binary weight is equal to their deficiency.
Numbers n such that A000120(n) = A033879(n), or equally A000203(n) = A005187(n), or equally A001065(n) = A011371(n).
2^(2^k-1) * (2^(2^k) + 1) is in the sequence if and only if (2^(2^k) + 1) is a (Fermat) prime (A019434) which as of today is known to be the case for 0 <= k <= 4 giving the terms 3, 10, 136, 32896 and 2147516416. - David A. Corneth, Nov 26 2017
It would be nice to know whether 315 is the only term that is neither in A191363 nor a power of two.
Any term that is either a square or twice a square (in A028982) must be odious (in A000069), and vice versa.
If there's an odd term below 10^30 besides 315 then it must be divisible by a prime >= 23. - David A. Corneth, Nov 27 2017
221753180448460815 is odd and also a term of this sequence. - Alexander Violette, Dec 24 2020

			A000203(315) = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 63 + 105 + 315 = 624. 2*315 - 624 = 6, and when 315 is written in binary, 100111011, we see that it has six 1-bits. Thus 315 is included in the sequence.

Max Alekseyev, Table of n, a(n) for n = 1..70 (first 52 terms from Giovanni Resta)
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Positions of zeros in A294898 and A294899.
Subsequence of A005100 and also of A295298.
Subsequences: A000079, A191363 (the five known terms).
Cf. A000120, A000203, A001065, A005187, A011371, A019434, A033879, A141548.

Select[Range[2^20], DivisorSigma[1, #] + DigitCount[#, 2, 1] == 2 # &] (* Michael De Vlieger, Nov 26 2017 *)

for(n=1, oo, if(sigma(n)+hammingweight(n) == 2*n, print1(n, ", ")));

Terms a(35) and beyond from Giovanni Resta, Feb 27 2020

A101223 Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.

Original entry on oeis.org

11, 21, 26, 68, 656, 2336, 8768, 133376, 528896, 34360918016, 35184409837568, 576460757135261696

Offset: 1

Vassil K. Tintschev (tinchev(AT)sunhe.jinr.ru), Dec 15 2004

a(13) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
A subsequence of A274556. a(11) <= b(23) = 35184409837568 ~ 3.5*10^13, since b(k) := 2^(k-1)*(2^k+9) is in this sequence for all k in A057196 (2^k+9 is prime). All known terms except a(2) = 21 are of that form. - M. F. Hasler, Jul 18 2016
Any term x of this sequence can be combined with any term y of A223609 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 divisors of 68 are {1, 2, 4, 17, 34, 68} and so sigma(68) = 1 + 2 + 4 + 17+ 24 + 68 = 126 = 2*68 - 10; thus, the deficiency of 68 is 10 so 68 is a term of the sequence.

Cf. A033879, A033880, A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A125248 (deficiency 16).
Cf. also A274556.
Cf. A223609 (abundance 10).

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

Select[ Range[ 85000000], DivisorSigma[1, # ] + 10 == 2# &]

Edited and extended by Robert G. Wilson v, Dec 15 2004
a(10) from Donovan Johnson, Dec 23 2008
Edited by M. F. Hasler, Jul 18 2016
a(11)-a(12) from Hiroaki Yamanouchi, Aug 21 2018

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

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A059242 Numbers k such that 2^k + 5 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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A191363 Numbers m such that sigma(m) = 2*m - 2.

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A125246 Numbers m whose abundance sigma(m) - 2m = -4. Numbers whose deficiency is 4.

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A087167 Odd numbers such that sigma(n) - 2n = 6.

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A295296 Numbers n such that the sum of their divisors + the number of ones in their binary expansion = 2n; numbers for which A000203(n) + A000120(n) = 2n.

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A101223 Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.

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