A050414 -id:A050414 - cp's OEIS frontend

A036563 a(n) = 2^n - 3.

-2, -1, 1, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 16381, 32765, 65533, 131069, 262141, 524285, 1048573, 2097149, 4194301, 8388605, 16777213, 33554429, 67108861, 134217725, 268435453, 536870909, 1073741821, 2147483645

Offset: 0

N. J. A. Sloane

a(n+1) is the n-th number with exactly n 1's in binary representation. - Reinhard Zumkeller, Mar 06 2003
Berstein and Onn: "For every m = 3k+1, the Graver complexity of the vertex-edge incidence matrix of the complete bipirtite graph K(3,m) satisfies g(m) >= 2^(k+2)-3." - Jonathan Vos Post, Sep 15 2007
Row sums of triangle A135857. - Gary W. Adamson, Dec 01 2007
a(n) = A164874(n-1,n-2) for n > 2. - Reinhard Zumkeller, Aug 29 2009
Starting (1, 5, 13, ...) = eigensequence of a triangle with A016777: (1, 4, 7, 10, ...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175655. For the central square just one A[5] vector, with decimal value 186, leads to this sequence (n >= 2). For the corner squares this vector leads to the companion sequence A123203. - Johannes W. Meijer, Aug 15 2010
First differences of A095264: A095264(n+1) - A095264(n) = a(n+2). - J. M. Bergot, May 13 2013
a(n+2) is given by the sum of n-th row of triangle of powers of 2: 1; 2 1 2; 4 2 1 2 4; 8 4 2 1 2 4 8; ... - Philippe Deléham, Feb 24 2014
Also, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508. - Robert Price, Mar 09 2017
a(n+3) is the value of the Ackermann function A(3,n) or ack(3,n). - Olivier Gérard, May 11 2018

			a(2) = 1;
a(3) = 2 + 1 + 2 = 5;
a(4) = 4 + 2 + 1 + 2 + 4 = 13;
a(5) = 8 + 4 + 2 + 1 + 2 + 4 + 8 = 29; etc. - _Philippe Deléham_, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Yael Berstein and Shmuel Onn, The Graver complexity of integer programming, Annals of Combinatorics, Vol. 13, No. 3 (2009), pp. 289-296; arXiv preprint, arXiv:0709.1500 [math.CO], 2007.
L' Education Mathématique, Problème 8907, 49e Annee, No 14, 15 Avril 1947, p. 113
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. Sequence is on page 267.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Row sums of triangular array A027960. A column of A119725.
Cf. A081118, A130459, A131112, A050414, A050415, A135857, A000079, A016777, A283508.
Cf. A074877, A304370, A304371, A331372.

List([0..40], n-> 2^n -3); # G. C. Greubel, Nov 18 2019

[2^n-3: n in [0..40]]; // Vincenzo Librandi, May 09 2011

A036563:=n->2^n-3; seq(A036563(n), n=0..40); # Wesley Ivan Hurt, Jun 26 2014

Table[2^n - 3, {n, 0, 40}] (* Wesley Ivan Hurt, Jun 26 2014 *)
LinearRecurrence[{3,-2},{-2,-1},40] (* Harvey P. Dale, Sep 26 2018 *)

a(n)= 2^n-3 \\ Charles R Greathouse IV, Dec 22 2011

def A036563(n): return (1<Chai Wah Wu, Sep 27 2024

[gaussian_binomial(n,1,2)-2 for n in range(0,40)] # Zerinvary Lajos, May 31 2009

a(n) = 2*a(n-1) + 3.
The sequence 1, 5, 13, ... has a(n) = 4*2^n-3. These are the partial sums of A151821. - Paul Barry, Aug 25 2003
a(n) = A118654(n-3, 6), for n > 2. - N. J. A. Sloane, Sep 29 2006
Row sums of triangle A130459 starting (1, 5, 13, 29, 61, ...). - Gary W. Adamson, May 26 2007
Row sums of triangle A131112. - Gary W. Adamson, Jun 15 2007
Binomial transform of [1, 4, 4, 4, ...] = (1, 5, 13, 29, 61, ...). - Gary W. Adamson, Sep 20 2007
a(n) = 2*StirlingS2(n,2) - 1, for n > 0. - Ross La Haye, Jul 05 2008
a(n) = A000079(n) - 3. - Omar E. Pol, Dec 21 2008
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-2*x) - 3/(1-x).
E.g.f.: exp(2*x) - 3*exp(x). (End)
For n >= 3, a(n) = 2<+>n, where operation <+> is defined in A206853. - Vladimir Shevelev, Feb 17 2012
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1, a(0)=-2, a(1)=-1. - Philippe Deléham, Dec 23 2013
Sum_{n>=1} 1/a(n) = A331372. - Amiram Eldar, Nov 18 2020

A057732 Numbers k such that 2^k + 3 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754, 38244, 39796, 40347, 55456, 58312, 122550, 205962, 235326, 363120, 479844, 685578, 742452, 1213815, 1434400, 1594947, 1875552, 1940812, 2205444

Offset: 1

G. L. Honaker, Jr., Oct 29 2000

Some of the larger entries may only correspond to probable primes.

A number k is in this sequence iff A062709(k) is in A057733; this is the case iff A257273(k) is in A125246. - M. F. Hasler, Apr 27 2015

			For k = 6, 2^6 + 3 = 67 is prime.
For k = 28, 2^28 + 3 = 268435459 is prime.

Mike Oakes, posting to primenumbers(AT)yahoogroups.com on Jul 08 2001

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+3, PRP Top Records.

Cf. A050414, A057733, A062709, A094076, A125246, A257273.

Cf. A019434 (primes 2^k+1), this sequence (2^k+3), A059242 (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).

[n: n in [0..1000] | IsPrime(2^n+3)]; // Vincenzo Librandi, Apr 27 2015

Select[Range[10000], PrimeQ[2^# + 3] &] (* Vincenzo Librandi, Apr 27 2015 *)

for(n=1, 2200, if(isprime(2^n+3), print1(n, ", ")));

for (n=1, 2, if (isprime(2^n+3), print1(n, ", "))); for(n=3, 100000, N=2^n+3 ; S=(N-5)/2 ; x=S ; for(j=1, n-1, x=Mod(x^2-2, N)) ; if(x==S , print1(n, ", "))) \\ produces terms corresponding to probable primes, see formula; Tony Reix, Aug 27 2015

Here is an LLT-like algorithm, using a cycle of the digraph x^2-2 modulo N, that finds terms of this sequence generating a PRP (PRobable Prime) of A057733 numbers: N=2^k+3; S0=(N-5)/2; s(0)=S0; s(i+1)=s(i)^2-2 modulo N; if s(k-1) == S0 then N is prime. - Tony Reix, Aug 27 2015

More terms from Jason Earls, Jul 18 2001 and Mike Oakes, Jul 28 2001
a(47)-a(50) from Donovan Johnson 2006, verified by Paul Bourdelais, Mar 22 2012
a(51) is a probable prime based on trial factoring to 1E9 and PRP testing base 3,5,7 (PFGW v3.3.1). Discovered by Paul Bourdelais, Apr 09 2012
a(52)-a(54) from Paul Bourdelais, Jun 18 2019
a(55) from Paul Bourdelais, Jul 16 2019
a(56) from Paul Bourdelais, Apr 22 2020
a(57) from Paul Bourdelais, Jun 12 2020
a(58) from Paul Bourdelais, Aug 04 2020

A045768 Numbers k such that sigma(k) == 2 (mod k).

Original entry on oeis.org

1, 20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Dan Hoey

nonn

Equivalently, Chowla function of k is congruent to 1 (mod k).
If p=2^i-3 is prime, then 2^(i-1)*p is a term of the sequence. 650 is in the sequence, but is not of this form.
Terms k from a(2) to a(14) satisfy sigma(k) = 2*k + 2, implying that sigma(k) == 0 (mod k+1). It is not known if this holds in general, for there might be solutions of sigma(k)=3k+2 or 4k+2 or ... (Comments from Jud McCranie and Dean Hickerson, updated by Jon E. Schoenfield, Sep 25 2021 and by Max Alekseyev, May 23 2025).
k | sigma(k) produces the multiperfect numbers (A007691). It is an open question whether k | sigma(k) - 1 iff k is a prime or 1. It is not known if there exist solutions to sigma(k) = 2k+1.
Sequence also gives the nonprime solutions to sigma(k) == 0 (mod k+1), k > 1. - Benoit Cloitre, Feb 05 2002
Sequence seems to give nonprime k such that the numerator of the sum of the reciprocals of the divisors of k equals k+1 (nonprime k such that A017665(k)=k+1). - Benoit Cloitre, Apr 04 2002
For k > 1, composite numbers k such that A108775(k) = floor(sigma(k)/k) = sigma(k) mod k = A054024(k). Complement of primes (A000040) with respect to A230606. There are no numbers k > 2 such that sigma(x) = k*(x+1) has a solution. - Jaroslav Krizek, Dec 05 2013

			sigma(650) = 1302 == 2 (mod 650).

R. K. Guy, Unsolved Problems in Number Theory, B2.

Amitabha Tripathi, A note on products of primes that differ by a fixed integer, Fibonacci Quart. 48 (2010), no. 2, 144-149.

Numbers k such that A054013(k)=1.

Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496.

Do[If[Mod[DivisorSigma[1, n]-2, n]==0, Print[n]], {n, 1, 10^8}]
Join[{1}, Select[Range[8000000], Mod[DivisorSigma[1, #], #]==2 &]] (* Vincenzo Librandi, Mar 11 2014 *)

is(n)=sigma(n)%n==2 || n==1 \\ Charles R Greathouse IV, Mar 09 2014

More terms from Jud McCranie, Dec 22 1999.
a(11) from Donovan Johnson, Mar 01 2012
a(12) from Giovanni Resta, Apr 02 2014
a(13) from Jud McCranie, Jun 02 2019
Edited and a(14) from Jon E. Schoenfield confirmed by Max Alekseyev, May 23 2025

A059608 Numbers k such that 2^k - 5 is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 18, 20, 26, 32, 36, 56, 66, 118, 130, 150, 166, 206, 226, 550, 706, 810, 1136, 1228, 1818, 2368, 2400, 3128, 4532, 5112, 8492, 16028, 16386, 17392, 18582, 21986, 24292, 27618, 30918, 32762, 48212, 120440, 183632, 316140, 364982, 414032, 533350, 595122

Offset: 1

Andrey V. Kulsha, Jan 30 2001

nonn

Except 3, all terms are even since for odd k, 2^k - 5 is divisible by 3.

			k = 10: 2^10 - 5 = 1019 is prime.
k = 20: 2^20 - 5 = 1048571 is prime.

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-5, PRP Top Records.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), this sequence (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

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

is(n)=isprime(2^n-5) \\ Charles R Greathouse IV, Feb 17 2017

a(32)-a(34) from Labos Elemer, Jul 09 2004
a(35)-a(40) from Max Alekseyev, a(41) from Paul Underwood, a(42)-a(46) from Henri Lifchitz, added by Max Alekseyev, Feb 09 2012
a(47)-a(48) from Jon Grantham, Jul 29 2023

A096818 Numbers k such that 2^k - 13 is prime.

Original entry on oeis.org

4, 5, 9, 13, 17, 57, 105, 137, 3217, 3229, 4233, 6097, 8757, 11457, 12073, 15425, 40117, 45357, 334809, 1509037

Offset: 1

Labos Elemer, Jul 13 2004

Except the first term 4, all terms are odd since for even k, 2^k - 13 is divisible by 3.

			k = 5: 32 - 13 = 19 is prime.

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-13, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), this sequence (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[4,20000],PrimeQ[2^#-13]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)

a(16) from Max Alekseyev, a(17)-a(18) from Henri Lifchitz, a(19) from Lelio R Paula, added by Max Alekseyev, Feb 09 2012

a(20) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 17 2023

A050415 Primes of the form 2^k - 3.

Original entry on oeis.org

5, 13, 29, 61, 509, 1021, 4093, 16381, 1048573, 4194301, 16777213, 536870909, 19807040628566084398385987581, 83076749736557242056487941267521533, 5316911983139663491615228241121378301, 1427247692705959881058285969449495136382746621

Offset: 1

Jud McCranie, Dec 22 1999

nonn

If p = 2^k - 3 is in this sequence, then p*2^(k-1) is abundant with abundance 2. - Claude Morin, Feb 01 2007

Equivalently, primes which give a prime number when 0's and 1's are interchanged in their binary representation; note that the resulting prime is always 10_2 = 2_10 (see A347476). - Bernard Schott, Nov 14 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..30
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.

Cf. A045768, A050414.

Subsequence of A347476.

[a: n in [1..200] | IsPrime(a) where a is 2^n-3]; // Vincenzo Librandi, Dec 08 2011

lst={};Do[p=2^n-3;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[2^n-3,{n,1,400}],PrimeQ] (* Vincenzo Librandi, Dec 08 2011 *)

for(n=3,999,if(ispseudoprime(t=2^n-3),print1(t", "))) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 2^A050414(n) - 3.

A059610 Numbers k such that 2^k - 9 is prime.

Original entry on oeis.org

4, 5, 9, 11, 17, 21, 33, 125, 141, 243, 251, 285, 321, 537, 563, 699, 729, 2841, 3365, 8451, 8577, 9699, 9725, 21011, 22689, 33921, 51761, 655845, 676761, 3480081

Offset: 1

Andrey V. Kulsha, Feb 02 2001

Except the first term 4, all terms are odd since 2^(2*m) - 9 = (2^m - 3)*(2^m + 3) is not prime for m > 2.

			243 is in the sequence because 2^243 - 9 is prime.

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. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), this sequence (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[3,20000],PrimeQ[2^#-9]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)

is(n)=isprime(2^n-9) \\ Charles R Greathouse IV, Feb 17 2017

a(24)-a(25) from Max Alekseyev, a(26)-a(27) from Paul Underwood, added by Max Alekseyev, Feb 09 2012
a(28)-a(29) from Robert Price, Jan 25 2017
a(30) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 17 2023

A059609 Numbers k such that 2^k - 7 is prime.

Original entry on oeis.org

39, 715, 1983, 2319, 2499, 3775, 12819, 63583, 121555, 121839, 468523, 908739

Offset: 1

Andrey V. Kulsha, Feb 02 2001

			k = 39, 2^39 - 7 = 549755813881 is prime.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 39, p. 15, Ellipses, Paris 2008.
J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 395 pp. 55; 218, Ellipses Paris 2004.
Wacław Sierpiński, Co wiemy, a czego nie wiemy o liczbach pierwszych. Warsaw: PZWS, 1961, pp. 46-47.
Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, pp. 31, 75.

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-7, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), this sequence (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[3, 20000], PrimeQ[2^# - 7] &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)

is(n)=isprime(2^n-7) \\ Charles R Greathouse IV, Feb 17 2017

a(8) from Henri Lifchitz, a(9)-a(10) from Gary Barnes, added by Max Alekseyev, Feb 09 2012
a(11) from Lelio R Paula, added by Max Alekseyev, Oct 25 2015
a(12) from Jon Grantham, Aug 09 2023

A059612 Numbers k such that 2^k - 15 is prime.

Original entry on oeis.org

5, 7, 8, 10, 14, 16, 23, 76, 95, 100, 158, 196, 235, 338, 620, 1646, 1850, 1891, 3833, 4394, 5194, 6017, 6070, 8824, 9955, 11399, 12250, 28723, 32057, 45494, 137359, 139627, 160654, 178819, 183284, 276391, 283466, 400571, 449030, 632815, 875518, 981016, 3511529

Offset: 1

Andrey V. Kulsha, Feb 13 2001

nonn

			100 is present because 2^100 - 15 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-15, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), this sequence (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

A059612:=n->if isprime(2^n-15) then n; fi; seq(A059612(n), n=1..20000); # Wesley Ivan Hurt, Dec 06 2013

Select[Range[4,20000],PrimeQ[2^#-15]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)

is(n)=isprime(2^n-15) \\ Charles R Greathouse IV, Feb 17 2017

a(26) from Labos Elemer, Jul 09 2004
a(27)-a(29) from Max Alekseyev, a(30) from Henri Lifchitz, a(31)-a(32) from Gary Barnes, a(33)-a(35) from Lelio R Paula, added by Max Alekseyev, Feb 09 2012
a(36)-a(37) from Lelio R Paula, added by Max Alekseyev, Oct 24 2013
a(38) from Lelio R Paula, added by Robert Price, Dec 06 2013
a(39) from Lelio R Paula, added by Robert Price, Mar 16 2019
a(40)-a(43) from Stefano Morozzi, added by Elmo R. Oliveira, Nov 16 2023

A088831 Numbers k whose abundance is 2: sigma(k) - 2k = 2.

Original entry on oeis.org

20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Labos Elemer, Oct 28 2003

A subset of A045768.
If 2^k-3 is prime (k is a term of A050414) then 2^(k-1)*(2^k-3) is in the sequence; this fact is a result of the following interesting theorem that I have found. Theorem: If j is an integer and 2^k-(2j+1) is prime then 2^(k-1)*(2^k-(2j+1)) is a solution of the equation sigma(x)=2(x+j). - Farideh Firoozbakht, Feb 23 2005
Note that the fact "if 2^p-1 is prime then 2^(p-1)*(2^p-1) is a perfect number" is also a trivial result of this theorem. All known terms of this sequence are of the form 2^(k-1)*(2^k-3) where 2^k-3 is prime. Conjecture: There are no terms of other forms. So the next terms of this sequence are likely 549754241024, 8796086730752, 140737463189504, 144115187270549504, 2^93*(2^94-3), 2^115*(2^116-3), 2^121*(2^122-3), 2^149*(2^150-3), etc. - Farideh Firoozbakht, Feb 23 2005
The conjecture in the previous comment is incorrect. The first counterexample is 650, which has factorization 2*5^2*13. - T. D. Noe, May 10 2010
a(11) > 10^12. - Donovan Johnson, Dec 08 2011
a(12) > 10^13. - Giovanni Resta, Mar 29 2013
a(14) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018
Any term x of this sequence can be combined with any term y of A191363 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 there any odd term in this sequence? - Jenaro Tomaszewski, Jan 06 2021
If there exists any odd term in this sequence, it must be weird, so it must exceed 10^28. - Alexander Violette, Jan 02 2022

			Abundances of terms in A045768: {-1,2,2,2,2,2,2,2,2,2} so 1 is not here.

Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 13, 1997.
Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Sec. B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.

P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.

Cf. A033880, A045768, A050414, A191363 (deficiency 2).

Select[Range[10^6], DivisorSigma[1, #] - 2 # == 2 &] (* Michael De Vlieger, Feb 25 2017 *)

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

Solutions to sigma(x)-2*x = 2.

One more term from Farideh Firoozbakht, Feb 23 2005
Comment and example corrected by T. D. Noe, May 10 2010
a(10) from Donovan Johnson, Dec 08 2011
a(11) from Giovanni Resta, Mar 29 2013
a(12) from Jud McCranie, Jun 18 2017
a(13) from Hiroaki Yamanouchi, Aug 23 2018

cp's OEIS Frontend

A036563 a(n) = 2^n - 3.

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A057732 Numbers k such that 2^k + 3 is prime.

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Magma

Mathematica

PARI

PARI

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Extensions

A045768 Numbers k such that sigma(k) == 2 (mod k).

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Mathematica

PARI

Extensions

A059608 Numbers k such that 2^k - 5 is prime.

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Mathematica

PARI

Extensions

A096818 Numbers k such that 2^k - 13 is prime.

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Mathematica

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A050415 Primes of the form 2^k - 3.

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Magma

Mathematica