A050415 -id:A050415 - 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

A050414 Numbers k such that 2^k - 3 is prime.

Original entry on oeis.org

3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 14114, 14400, 16460, 16680, 20757, 26350, 30041, 34452, 36552, 42689, 44629, 50474, 66422, 69337, 116926, 119324, 123297, 189110, 241004, 247165, 284133, 354946, 394034, 702194, 750740, 840797, 1126380, 1215889, 1347744, 1762004, 2086750

Offset: 1

Jud McCranie, Dec 22 1999

nonn

With 65 known primes corresponding to k < 1762005, these primes appear to be more common than Mersenne primes.  Of course at this time, the larger terms correspond only to probable primes. - Paul Bourdelais, Feb 04 2012
The numbers 2^k-3 and 2^k-1 are both primes for k = 3, 5, ? The lesser number 2^p-3 is prime for primes p = 3, 5, 29, 233, 42689, 69337, ... (see A283266). - Thomas Ordowski, Sep 18 2015
The terms a(43)-a(49) were found by Paul Underwood, a(50)-a(51) found by M. Frind and P. Underwood, a(52) found by Gary Barnes, a(53)-a(58) found by M. Frind and P. Underwood, and a(59)-a(66) found by Paul Bourdelais (see link Henri Lifchitz and Renaud Lifchitz). - Elmo R. Oliveira, Dec 02 2023

			k = 22, 2^22 - 3 = 4194301 is prime.
k = 24, 2^24 - 3 = 16777213 is prime.

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. A045768, A050415, A057732 (numbers k such that 2^k + 3 is prime).
For prime terms see A283266.
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), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]

for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014

More terms from Robert G. Wilson v, Sep 15 2000
More terms from Andrey V. Kulsha, Feb 11 2001
a(40) verified with 20 iterations of Miller-Rabin test, from Dmitry Kamenetsky, Jul 12 2008
a(41) a new PRP term, from Serge Batalov, Oct 20 2008
Corrected and extended by including two smaller (apparently known) PRP and 16 larger terms from PRP Top Records of this form, all discovered by M. Frind & P. Underwood, Gary Barnes, Oct 20 2008
a(59)-a(60) discovered by Paul Bourdelais, Mar 26 2012
a(61)-a(63) discovered by Paul Bourdelais, Jun 18 2019
a(64) discovered by Paul Bourdelais, Jul 16 2019
a(65) discovered by Paul Bourdelais, Apr 20 2020
a(66) discovered by Paul Bourdelais, May 28 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

A234451 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k)/2 + phi(m)/6) + 3 is prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 3, 5, 4, 5, 4, 6, 4, 4, 5, 5, 5, 6, 6, 6, 5, 6, 8, 7, 6, 5, 7, 8, 7, 10, 6, 7, 9, 7, 5, 5, 8, 6, 6, 7, 9, 3, 7, 10, 9, 3, 8, 6, 8, 6, 9, 9, 12, 5, 8, 8, 10, 9, 10, 9, 8, 8, 8, 10, 9, 12, 10, 13, 11, 9, 10

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 9. Also, any integer n > 13 can be written as k + m with k > 0 and m > 0 such that 2^(phi(k)/2 + phi(m)/6) - 3 is prime.
(ii) Each integer n > 25 can be written as k + m with k > 0 and m > 0 such that 3*2^(phi(k)/2 + phi(m)/8) + 1 (or 3*2^(phi(k)/2 + phi(m)/12) + 1 when n > 38) is prime. Also, any integer n > 14 can be written as k + m with k > 0 and m > 0 such that 3*2^(phi(k)/2 + phi(m)/12) - 1 is prime.
This conjecture implies that there are infinitely many primes in any of the four forms 2^n + 3, 2^n - 3, 3*2^n + 1, 3*2^n - 1.
We have verified the conjecture for n up to 50000.

			a(10) = 1 since 10 = 3 + 7 with 2^(phi(3)/2 + phi(7)/6) + 3 = 7 prime.
a(11) = 1 since 11 = 4 + 7 with 2^(phi(4)/2 + phi(7)/6) + 3 = 7 prime.
a(12) = 2 since 12 = 3 + 9 = 5 + 7 with 2^(phi(3)/2 + phi(9)/6) + 3 = 7 and 2^(phi(5)/2 + phi(7)/6) + 3 = 11 both prime.
a(769) = 1 since 769 = 31 + 738 with 2^(phi(31)/2 + phi(738)/6) + 3 = 2^(55) + 3 prime.
a(787) = 1 since 787 = 112 + 675 with 2^(phi(112)/2 + phi(675)/6) + 3 = 2^(84) + 3 prime.
a(867) = 1 since 867 = 90 + 777 with 2^(phi(90)/2 + phi(777)/6) + 3 = 2^(84) + 3 prime.
a(869) = 1 since 869 = 51 + 818 with 2^(phi(51)/2 + phi(818)/6) + 3 = 2^(84) + 3 prime.
a(913) = 1 since 913 = 409 + 504 with 2^(phi(409)/2 + phi(504)/6) + 3 = 2^(228) + 3 prime.
a(1085) = 1 since 1085 = 515 + 570 with 2^(phi(515)/2 + phi(570)/6) + 3 = 2^(228) + 3 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000079, A050415, A057733, A234309, A234310, A234337, A234344, A234346, A234347, A234359, A234360, A234361, A234388, A234399, A234470, A236358.

f[n_,k_]:=2^(EulerPhi[k]/2+EulerPhi[n-k]/6)+3
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A135535 Primes of the form 4^k - 3.

Original entry on oeis.org

13, 61, 1021, 4093, 16381, 1048573, 4194301, 16777213, 19807040628566084398385987581, 83076749736557242056487941267521533, 5316911983139663491615228241121378301, 1427247692705959881058285969449495136382746621, 23945242826029513411849172299223580994042798784118781, 118571099379011784113736688648896417641748464297615937576404566024103044751294461, 139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444733

Offset: 1

Daniele Corradetti (d.corradetti(AT)gmail.com), Feb 21 2008

nonn

Involved in the "New Mersenne Prime Conjecture" and in some generalizations of Mersenne primes.

Subsequence of A050415. - Elmo R. Oliveira, Nov 28 2023

			16381 is a term because 4^7 - 3 = 16381 is prime.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (pp. 114-134).

P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., The New Mersenne Conjecture, Amer. Math. Monthly 96, 125-128, 1989.
D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Paul Tannery, Questions 659 et 660, L'Intermédiaire des mathématiciens, Tome II (1895) p. 317.
Eric Weisstein's World of Mathematics, New Mersenne Prime Conjecture

Cf. A000040, A050415, A057732, A057733, A059266.

Do[If[PrimeQ[4^n - 3], Print[4^n - 3]], {n, 100}] (* Robert G. Wilson v, Feb 29 2008 *)
Select[4^Range[200]-3,PrimeQ] (* Harvey P. Dale, Jul 11 2022 *)

a(n) = 4^A059266(n) - 3. - Ryan Propper, Feb 26 2008

More terms from R. J. Mathar, Robert G. Wilson v and Ryan Propper, Feb 26 2008

A181703 Numbers of the form 2^(t-1)*(2^t-3), where 2^t-3 is prime.

Original entry on oeis.org

20, 104, 464, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504, 196159429230833773869868419445529014560349481041922097152, 3450873173395281893717377931138512601610429881249330192849350210617344

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

This is a subsequence of A181595. [Proof: sigma(m) = (2^t-1)*(2^t-2) leads to an abundance of m which is 2.]
Numbers m such that the sum of the even divisors of m equals the square of the odd divisors of m.
Proof: let s0 the sum of the even divisors and s1 the sum of the odd divisors.
s1 = 2^t-2 because 2^t-3 is prime.
s0 = 2 + 4 + 8 + ... + 2^(t-1) + (2^t - 3)(2 + 4 + 8 + ... + 2^(t-1)) = (2^t - 2)^2 => s0 = s1^2. - Michel Lagneau, Apr 17 2013

Eric Chen, Table of n, a(n) for n = 1..28
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.

Cf. A181595, A181701, A000396, A050414, A050415.

with(numtheory):for n from 1 to 600000 do:x:=divisors(n):n0:=nops(x):s0:=0:s1:=0:for k from 1 to n0 do:if irem(x[k],2)=0 then s0:=s0+ x[k]:else s1:=s1+ x[k]:fi:od:if s0=s1^2 then print(n):else fi:od: # Michel Lagneau, Apr 17 2013

for(k=1, 200, if(ispseudoprime(2^k-3), print1(2^(k-1)*(2^k-3), ", "))) \\ Eric Chen, Jun 13 2018

a(n) = 2^(A050414(n)-1) * (2^A050414(n) - 3). - Max Alekseyev, Jul 31 2025

Edited and extended by D. S. McNeil, Nov 18 2010

Definition simplified by R. J. Mathar, Nov 18 2010

A235490 Numbers such that none of their prime factors share common 1-bits in the same bit-position and when added (or "ored" or "xored") together, yield a term of A000225 (a binary "repunit").

Original entry on oeis.org

1, 3, 7, 10, 26, 31, 58, 122, 127, 1018, 2042, 8186, 8191, 32762, 131071, 524287, 2097146, 8388602, 33554426, 1073741818, 2147483647, 2305843009213693951, 618970019642690137449562111, 39614081257132168796771975162, 162259276829213363391578010288127, 166153499473114484112975882535043066

Offset: 1

Antti Karttunen, Jan 22 2014

a(1) = 1 is included on the grounds that it has no prime factors, thus A001414(1)=0, and 0 is one of the terms of A000225, marking the "repunit of length zero".

After 1, the sequence is a union of A000668 (Mersenne primes) and semiprimes of the form 2*A050415. The terms were constructed from the data given in those two entries.

			7 is included, because it is a prime, and repunit in base-2: '111'.
10 is included, as 10=2*5, and when we add 2 ('10' in binary) and 5 ('101' in binary), we also get 7 ('111' in binary), without producing any carries.

Subsequence of A235050.
Cf. A000225, A000668, A050415, A001414, A072593, A178910, A227843.
Cf. also A235488, A230492, A235032, A050376.

A225582 Primes in the chain of repeated application of x->2*x+3, starting at x=11.

Original entry on oeis.org

11, 53, 109, 1789, 3581, 28669, 229373, 3670013, 58720253, 117440509, 60129542141, 264452523040700131966973, 34662321099990647697175478269, 2381976568446569244243622252022377480189, 4878288012178573812210938372141829079433213

Offset: 1

Vincenzo Librandi, May 14 2013

Primes in A156127.

Vincenzo Librandi, Table of n, a(n) for n = 1..26

Primes of the form k*2^m-3: A050415 (k=1), A173769 (k=5), this sequence (k=7), A225583 (k=11), A225584 (k=13); A172156 (k=100), A163589 (k=715).

Numbers of the form 7*2^m-3: A156127.

x:=11; a:=[n eq 1 select x else 2*Self(n-1)+3: n in [1..200]]; [a[i]: i in [1..#a] | IsPrime(a[i])];

[a: n in [0..150] | IsPrime(a) where a is 7*2^n-3];

Select[NestList[2 # + 3 &, 11, 30], PrimeQ] (* or *) Select[Table[7 2^n - 3, {n, 0, 150}], PrimeQ]

A225583 Primes in the chain of repeated application of x->2*x+3, starting at x=19.

Original entry on oeis.org

19, 41, 173, 349, 701, 11261, 45053, 180221, 184549373, 738197501, 5905580029, 188978561021, 6192449487634429, 49539595901075453, 99079191802150909, 25364273101350633469, 811656739243220271101, 3486039150627630854115933814781

Offset: 1

Vincenzo Librandi, May 14 2013

nonn

Primes in A156198.

Vincenzo Librandi, Table of n, a(n) for n = 1..37

Primes of the form k*2^m-3: A050415 (k=1), A173769 (k=5), A225582 (k=7), this sequence (k=11), A225584 (k=13); A172156 (k=100), A163589 (k=715).

Numbers of the form 11*2^m-3: A156198.

x:=19; a:=[n eq 1 select x else 2*Self(n-1)+3: n in [1..200]]; [a[i]: i in [1..#a] | IsPrime(a[i])];

[a: n in [0..150] | IsPrime(a) where a is 11*2^n-3];

Select[NestList[2 # + 3 &, 19, 30], PrimeQ] (* or *) Select[Table[11 2^n - 3, {n, 0, 150}], PrimeQ]

A225584 Primes in the chain of repeated application of x->2*x+3, starting at x=23.

Original entry on oeis.org

23, 101, 829, 6653, 13309, 6815741, 436207613, 13958643709, 27917287421, 468374361246531581, 3746994889972252669, 959230691832896684029, 16093220510709943573688614909

Offset: 1

Vincenzo Librandi, May 14 2013

Primes in A156202.

Vincenzo Librandi, Table of n, a(n) for n = 1..31

Primes of the form k*2^m-3: A050415 (k=1), A173769 (k=5), A225582 (k=7), A225583 (k=11), this sequence (k=13); A172156 (k=100), A163589 (k=715).

Numbers of the form 13*2^m-3: A156202.

x:=23; a:=[n eq 1 select x else 2*Self(n-1)+3: n in [1..200]]; [a[i]: i in [1..#a] | IsPrime(a[i])];

[a: n in [0..150] | IsPrime(a) where a is 13*2^n-3];

Select[NestList[2 # + 3 &, 23, 30], PrimeQ] (* or *) Select[Table[13 2^(n+1) - 3, {n, 0, 150}], PrimeQ]

cp's OEIS Frontend

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

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A050414 Numbers k such that 2^k - 3 is prime.

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

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A234451 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k)/2 + phi(m)/6) + 3 is prime, where phi(.) is Euler's totient function.

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A135535 Primes of the form 4^k - 3.

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A181703 Numbers of the form 2^(t-1)*(2^t-3), where 2^t-3 is prime.

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A235490 Numbers such that none of their prime factors share common 1-bits in the same bit-position and when added (or "ored" or "xored") together, yield a term of A000225 (a binary "repunit").

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