A077585 -id:A077585 - cp's OEIS frontend

A000215 Fermat numbers: a(n) = 2^(2^n) + 1.

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937

Offset: 0

N. J. A. Sloane

It is conjectured that just the first 5 numbers in this sequence are primes.
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004
For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether n is even or odd (Koshy). - Lekraj Beedassy, Mar 17 2005
This is the special case k=2 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058. - Seppo Mustonen, Sep 04 2005
For n>1 final two digits of a(n) are periodically repeated with period 4: {17, 57, 37, 97}. - Alexander Adamchuk, Apr 07 2007
For 1 < k <= 2^n, a(A007814(k-1)) divides a(n) + 2^k. More generally, for any number k, let r = k mod 2^n and suppose r != 1, then a(A007814(r-1)) divides a(n) + 2^k. - T. D. Noe, Jul 12 2007
From Daniel Forgues, Jun 20 2011: (Start)
The Fermat numbers F_n are F_n(a,b) = a^(2^n) + b^(2^n) with a = 2 and b = 1.
For n >= 2, all factors of F_n = 2^(2^n) + 1 are of the form k*(2^(n+2)) + 1 (k >= 1).
The products of distinct Fermat numbers (in their binary representation, see A080176) give rows of Sierpiński's triangle (A006943). (End)
Let F(n) be a Fermat number. For n > 2, F(n) is prime if and only if 5^((F(n)-1)/4) == sqrt(F(n)-1) (mod F(n)). - Arkadiusz Wesolowski, Jul 16 2011
Conjecture: let the smallest prime factor of Fermat number F(n) be P(F(n)). If F(n) is composite, then P(F(n)) < 3*2^(2^n/2 - n - 2). - Arkadiusz Wesolowski, Aug 10 2012
The Fermat primes are not Brazilian numbers, so they belong to A220627, but the Fermat composites are Brazilian numbers so they belong to A220571. For a proof, see Proposition 3 page 36 on "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
It appears that this sequence is generated by starting with a(0)=3 and following the rule "Write in binary and read in base 4".  For an example of "Write in binary and read in ternary", see A014118. - John W. Layman, Jul 30 2013
Conjecture: the numbers > 5 in this sequence, i.e., 2^2^k + 1 for k>1, are exactly the numbers n such that (n-1)^4-1 divides 2^(n-1)-1. - M. F. Hasler, Jul 24 2015

			a(0) = 1*2^1 + 1 = 3 = 1*(2*1) + 1.
a(1) = 1*2^2 + 1 = 5 = 1*(2*2) + 1.
a(2) = 1*2^4 + 1 = 17 = 2*(2*4) + 1.
a(3) = 1*2^8 + 1 = 257 = 16*(2*8) + 1.
a(4) = 1*2^16 + 1 = 65537 = 2048*(2*16) + 1.
a(5) = 1*2^32 + 1 = 4294967297 = 641*6700417 = (10*(2*32) + 1)*(104694*(2*32) + 1).
a(6) = 1*2^64 + 1 = 18446744073709551617 = 274177*67280421310721 = (2142*(2*64) + 1)*(525628291490*(2*64) + 1).

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7.
P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 87.
James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
R. K. Guy, Unsolved Problems in Number Theory, A3.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14.
E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.
T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational Mathematics Vol. 32 No. 2 2002-3 Baywood NY.
M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 36.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 18, 59.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 202.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 6-7, 70-75.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 148-149.

N. J. A. Sloane, Table of n, a(n) for n = 0..11
Richard Bellman, A note on relatively prime sequences, Bull. Amer. Math. Soc. 53 (1947), 778-779.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Chris K. Caldwell, The Prime Glossary, Fermat number.
Chris K. Caldwell, The prime pages All prime-square Mersenne divisors are Wieferich (2021).
Shane Chern, Fermat Numbers in Multinomial Coefficients, J. Int. Seq. 17 (2014) # 14.3.2.
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
Leonhard Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus.
Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
Jiří Klaška, Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
T.-W. Leung, A Brief Introduction to Fermat Numbers
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Michael A. Morrison and John Brillhart, The factorization of F_7, Bull. Amer. Math. Soc. 77 (1971), 264.
Robert Munafo, Fermat Numbers
Robert Munafo, Notes on Fermat numbers
Seppo Mustonen, On integer sequences with mutual k-residues.
Seppo Mustonen, On integer sequences with mutual k-residues. [Local copy]
OEIS Wiki, Fermat numbers.
OEIS Wiki, Sierpinski's triangle.
G. A. Paxson, The compositeness of the thirteenth Fermat number, Math. Comp. 15 (76) (1961) 420-420.
Carl Pomerance, A tale of two sieves, Notices Amer. Math. Soc., 43 (1996), 1473-1485.
P. Sanchez, PlanetMath.org, Fermat Numbers
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38. Local copy, included here with permission from the editors of Quadrature.
G. Villemin's Almanach of Numbers, Nombres de Fermat.
Le Roy J. Warren, Henry G. Bray, On the square-freeness of Fermat and Mersenne Numbers, Pac. J. Math. 22 (3) (1967) 563.
Eric Weisstein's World of Mathematics, Fermat Number.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
Wikipedia, Fermat number.
Wolfram Research, Fermat numbers are pairwise coprime.

a(n) = A001146(n) + 1 = A051179(n) + 2.
Cf. A019434, A050922, A051158, A051179, A063486, A073617, A085866.
See A004249 for a similar sequence.
Cf. A080176 for binary representation of Fermat numbers.
Cf. A220627, A220570, A220571, A125134, A249119.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.

a000215 = (+ 1) . (2 ^) . (2 ^)  -- Reinhard Zumkeller, Feb 13 2015

Maple
```
A000215 := n->2^(2^n)+1;
```

Table[2^(2^n) + 1, {n, 0, 8}] (* Alonso del Arte, Jun 07 2011 *)

A000215(n):=2^(2^n)+1$ makelist(A000215(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

PARI
```
a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
```

def a(n): return 2**(2**n) + 1
print([a(n) for n in range(9)]) # Michael S. Branicky, Apr 19 2021

a(0) = 3; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(n) = a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get the empty product, i.e., 1, plus 2, giving 3 = a(0). - Benoit Cloitre, Sep 15 2002 [edited by Daniel Forgues, Jun 20 2011]
The above formula implies that the Fermat numbers (being all odd) are coprime.
Conjecture: F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - Benoit Cloitre, Sep 15 2002
A000120(a(n)) = 2. - Reinhard Zumkeller, Aug 07 2010
If a(n) is composite, then a(n) = A242619(n)^2 + A242620(n)^2 = A257916(n)^2 - A257917(n)^2. - Arkadiusz Wesolowski, May 13 2015
Sum_{n>=0} 1/a(n) = A051158. - Amiram Eldar, Oct 27 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = A249119.
Product_{n>=0} (1 - 1/a(n)) = 1/2. (End)
a(n) = 2*A077585(n) + 3. - César Aguilera, Jul 26 2023
a(n) = 2*2^A000225(n) + 1. - César Aguilera, Jul 11 2024

A058891 a(n) = 2^(2^(n-1) - 1).

Original entry on oeis.org

1, 2, 8, 128, 32768, 2147483648, 9223372036854775808, 170141183460469231731687303715884105728, 57896044618658097711785492504343953926634992332820282019728792003956564819968

Offset: 1

N. J. A. Sloane, Jan 08 2001

For n > 1, a(n) is the least solution > 1 to rad(x)^(n-1) = tau(x) where rad(x) = A007947(x) is the squarefree kernel of x and tau(x) = A000005(x) the number of divisors of x. - Benoit Cloitre, Apr 18 2002 [Corrected by Michel Marcus, Oct 15 2018]
For n > 1, a(n) is also the total number of possible outcomes of a knockout tournament starting with 2^(n-1) players, taking account of all matches in the tournament. - Martin Griffiths, Mar 26 2009
Also, a(n+1) = 2^(2^n-1) for n >= 1 are solutions x = y of the Diophantine equation x^y * y^x = (x+y)^z in positive integers; corresponding solutions z are in A348332 (see this last sequence for more informations and links). - Bernard Schott, Oct 13 2021
For n > 2, a(n) ends with 8. - Bernard Schott, Oct 20 2021
a(n) is the number of labeled hypergraphs on n - 1 vertices. - Lorenzo Sauras Altuzarra, Apr 01 2023

			The 8 possible hyperedge sets for the vertex set {1, 2} are {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {2}}, {{1}, {1, 2}}, {{2}, {1, 2}} and {{1}, {2}, {1, 2}}. - _Lorenzo Sauras Altuzarra_, Apr 01 2023

F. Harary, Graph Theory, Page 209, Problem 16.11.

Harry J. Smith, Table of n, a(n) for n = 1..12
Wikipedia, Hypergraph.

Cf. A000079, A000225, A053287, A076214, A077585, A348332.

a[1]:=1: for n from 2 to 20 do a[n]:=2*a[n-1]^2 od: seq(a[n], n=1..9); # Zerinvary Lajos, Apr 16 2009

a = 1; b = -3; Table[Expand[(-1/2) ((a + Sqrt[b])^(2^n) + (a - Sqrt[b])^(2^n))], {n, 1, 10}] (* Artur Jasinski, Oct 11 2008 *)

a(n) = { 2^(2^(n-1)-1) } \\ Harry J. Smith, Jun 23 2009

def A058891(n): return 1<<(1<Chai Wah Wu, Dec 12 2022

a(n) = A053287(A000079(n-1)).
a(1) = 1, a(n+1) = 2*a(n)^2.
a(1) = 1, a(n+1) = 2^n*a(1)*a(2)*...*a(n). - Benoit Cloitre, Sep 13 2003
a(n) = (-1/2)*((1 + sqrt(-3))^(2^n) + (1 - sqrt(-3))^(2^n)). - Artur Jasinski, Oct 11 2008
a(n) = 2*a(n-1)^2 is an example with a(1) = 1 and k = 2 of a(n) = k*a(n-1)^2; general explicit formula: a(n) = ((a(1)*k)^(2^(n-1)))/k. - Andreas Pfaffel (andreas.pfaffel(AT)gmx.at), Apr 27 2010
a(n) = A077585(n-1) + 1. - Maurizio De Leo, Feb 25 2015
a(n) = 2^A000225(n-1). - Michel Marcus, Aug 19 2020
Sum_{n>=0} 1/a(n) = A076214. - Amiram Eldar, Oct 27 2020

A056220 a(n) = 2*n^2 - 1.

Original entry on oeis.org

-1, 1, 7, 17, 31, 49, 71, 97, 127, 161, 199, 241, 287, 337, 391, 449, 511, 577, 647, 721, 799, 881, 967, 1057, 1151, 1249, 1351, 1457, 1567, 1681, 1799, 1921, 2047, 2177, 2311, 2449, 2591, 2737, 2887, 3041, 3199, 3361, 3527, 3697, 3871, 4049, 4231, 4417, 4607, 4801

Offset: 0

N. J. A. Sloane, Aug 06 2000

Image of squares (A000290) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}. - Henry Bottomley, Dec 12 2000
Surround numbers of an n X n square. - Jason Earls, Apr 16 2001
Numbers n such that 2*n + 2 is a perfect square. - Cino Hilliard, Dec 18 2003, Juri-Stepan Gerasimov, Apr 09 2016
The sums of the consecutive integer sequences 2n^2 to 2(n+1)^2-1 are cubes, as 2n^2 + ... + 2(n+1)^2-1 = (1/2)(2(n+1)^2 - 1 - 2n^2 + 1)(2(n+1)^2 - 1 + 2n^2) = (2n+1)^3. E.g., 2+3+4+5+6+7 = 27 = 3^3, then 8+9+10+...+17 = 125 = 5^3. - Andras Erszegi (erszegi.andras(AT)chello.hu), Apr 29 2005
X values (other than 0) of solutions to the equation 2*X^3 + 2*X^2 = Y^2. To find Y values: b(n) = 2n*(2*n^2 - 1). - Mohamed Bouhamida, Nov 06 2007
Average of the squares of two consecutive terms is also a square. In fact: (2*n^2 - 1)^2 + (2*(n+1)^2 - 1)^2 = 2*(2*n^2 + 2*n + 1)^2. - Matias Saucedo (solomatias(AT)yahoo.com.ar), Aug 18 2008
Equals row sums of triangle A143593 and binomial transform of [1, 6, 4, 0, 0, 0, ...] with n > 1. - Gary W. Adamson, Aug 26 2008
Start a spiral of square tiles. Trivially the first tile fits in a 1 X 1 square. 7 tiles fit in a 3 X 3 square, 17 tiles fit in a 5 X 5 square and so on. - Juhani Heino, Dec 13 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-2, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 26 2010
For each n > 0, the recursive series, formula S(b) = 6*S(b-1) - S(b-2) - 2*a(n) with S(0) = 4n^2-4n+1 and S(1) = 2n^2, has the property that every even term is a perfect square and every odd term is twice a perfect square. - Kenneth J Ramsey, Jul 18 2010
Fourth diagonal of A154685 for n > 2. - Vincenzo Librandi, Aug 07 2010
First integer of (2*n)^2 consecutive integers, where the last integer is 3 times the first + 1. As example, n = 2: term = 7; (2*n)^2 = 16; 7, 8, 9, ..., 20, 21, 22: 7*3 + 1 = 22. - Denis Borris, Nov 18 2012
Chebyshev polynomial of the first kind T(2,n). - Vincenzo Librandi, May 30 2014
For n > 0, number of possible positions of a 1 X 2 rectangle in a (n+1) X (n+2) rectangular integer lattice. - Andres Cicuttin, Apr 07 2016
This sequence also represents the best solution for Ripà's n_1 X n_2 X n_3 dots problem, for any 0 < n_1 = n_2 < n_3 = floor((3/2)*(n_1 - 1)) + 1. - Marco Ripà, Jul 23 2018

			a(0) = 0^2-1*1 = -1, a(1) = 1^2 - 4*0 = 1, a(2) = 2^2 - 9*1 = 7, etc.
a(4) = 31 = (1, 3, 3, 1) dot (1, 6, 4, 0) = (1 + 18 + 12 + 0). - _Gary W. Adamson_, Aug 29 2008

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq., Vol. 13 (2010), Article # 10.7.8.
Mitch Phillipson, Manda Riehl, and Tristan Williams, Enumeration of Wilf classes in Sn ~ Cr for two patterns of length 3, PU. M. A., Vol. 21, No. 2 (2010), pp. 321-338.
Marco Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem , Notes on Number Theory and Discrete Mathematics, Vol. 20, No. 1 (2014), pp. 59-71.
Leo Tavares, Illustration: Twin Squares
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000105, A000217, A000290, A000384, A001082, A001653, A001844, A002378, A002593, A003215, A005563, A028347, A036666, A046092, A047875, A062717, A069074, A077585, A087475, A119258, A143593, A154685, A162610, A188653, A225227.
Cf. A066049 (indices of prime terms)
Column 2 of array A188644 (starting at offset 1).

List([0..50], n-> 2*n^2-1); # Muniru A Asiru, Jul 24 2018

[2*n^2-1 : n in [0..50]]; // Vincenzo Librandi, May 30 2014

A056220:=n->2*n^2-1; seq(A056220(n), n=0..50); # Wesley Ivan Hurt, Jun 16 2014

Array[2 #^2 - 1 &, 50, 0] (* Robert G. Wilson v, Jul 23 2018 *)
CoefficientList[Series[(x^2 +4x -1)/(1-x)^3, {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {-1, 1, 7}, 51] (* Robert G. Wilson v, Jul 24 2018 *)

PARI
```
a(n)=2*n^2-1;
```

[2*n^2-1 for n in (0..50)] # G. C. Greubel, Jul 07 2019

G.f.: (-1 + 4*x + x^2)/(1-x)^3. - Henry Bottomley, Dec 12 2000
a(n) = A119258(n+1,2) for n > 0. - Reinhard Zumkeller, May 11 2006
From Doug Bell, Mar 08 2009: (Start)
a(0) = -1,
a(n) = sqrt(A001844(n)^2 - A069074(n-1)),
a(n+1) = sqrt(A001844(n)^2 + A069074(n-1)) = sqrt(a(n)^2 + A069074(n-1)*2). (End)
a(n) + a(n+1) + 1 = (2n+1)^2. - Doug Bell, Mar 09 2009
a(n) = a(n-1) + 4*n - 2 (with a(0)=-1). - Vincenzo Librandi, Dec 25 2010
a(n) = A188653(2*n) for n > 0. - Reinhard Zumkeller, Apr 13 2011
a(n) = A162610(2*n-1,n) for n > 0. - Reinhard Zumkeller, Jan 19 2013
a(n) = ( Sum_{k=0..2} (C(n+k,3)-C(n+k-1,3))*(C(n+k,3)+C(n+k+1,3)) ) - (C(n+2,3)-C(n-1,3))*(C(n,3)+C(n+3,3)), for n > 3. - J. M. Bergot, Jun 16 2014
a(n) = j^2 + k^2 - 2 or 2*j*k if n >= 2 and j = n + sqrt(2)/2 and k = n - sqrt(2)/2. - Avi Friedlich, Mar 30 2015
a(n) = A002593(n)/n^2. - Bruce J. Nicholson, Apr 03 2017
a(n) = A000384(n) + n - 1. - Bruce J. Nicholson, Nov 12 2017
a(n)*a(n+k) + 2k^2 = m^2 (a perfect square), m = a(n) + (2n*k), for n>=1. - Ezhilarasu Velayutham, May 13 2019
From Amiram Eldar, Aug 10 2020: (Start)
Sum_{n>=1} 1/a(n) = 1/2 - sqrt(2)*Pi*cot(Pi/sqrt(2))/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi*csc(Pi/sqrt(2))/4 - 1/2. (End)
From Amiram Eldar, Feb 04 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(2))*csc(Pi/sqrt(2)).
Product_{n>=2} (1 - 1/a(n)) = (Pi/(4*sqrt(2)))*csc(Pi/sqrt(2)). (End)
a(n) = A003215(n) - A000217(n-2)*2. - Leo Tavares, Jun 29 2021
Let T(n) = n*(n+1)/2. Then a(n)^2 = T(2n-2)*T(2n+1) + n^2. - Charlie Marion, Feb 12 2023
E.g.f.: exp(x)*(2*x^2 + 2*x - 1). - Stefano Spezia, Jul 08 2023

A077586 a(n) = 2^(2^prime(n) - 1) - 1.

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727

Offset: 1

Henry Bottomley, Nov 07 2002

nonn

First four terms are primes. Fifth (1.61585...*10^616), sixth (5.45374...*10^2465), seventh (2.007...*10^39456) and eighth (1.298...*10^157826) are not primes.
Note that a(n) divides 2^a(n)-2 for every n, so if a(n) is composite then a(n) is a Fermat pseudoprime to base 2; cf. A007013. - Thomas Ordowski, Apr 08 2016
A number MM(p) is prime iff M(p) = A000225(p) = 2^p-1 is a Mersenne prime exponent (A000043), which isn't possible unless p itself is also in A000043. Primes of this form are called double Mersenne primes MM(p). For all Mersenne exponents between 7 and 61, factors of MM(p) are known. The next candidate MM(61) is far too large to be merely stored on any existing hard drive (it would require 3*10^17 bytes), but a distributed search for factors of this and other MM(p) is ongoing, see the doublemersenne.org web site. - M. F. Hasler, Mar 05 2020

			a(3) = 2^(2^5 - 1) - 1 = 2^31 - 1 = 2147483647.

Michael De Vlieger, Table of n, a(n) for n = 1..5
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Luigi Morelli, DoubleMersennes.org
Eric Weisstein's World of Mathematics, Double Mersenne Number
Wikipedia, Double Mersenne number

Cf. A077585 (double Mersenne numbers), A000225 (Mersenne numbers), A001348 (ditto with prime indices), A000040 (primes).

A077586 := n -> 2^(2^ithprime(n)-1)-1; A077586(n) $ n=1..5; # M. F. Hasler, Mar 05 2020

Array[2^(2^Prime[#] - 1) - 1 &, 4] (* Michael De Vlieger, Apr 14 2022 *)

apply( {A077586(n)=2^(2^prime(n)-1)-1}, [1..5]) \\ M. F. Hasler, Mar 05 2020

a(n) = A077585(A000040(n)) = A000225(A001348(n)).

A070969 a(n) = 2^(2^n + 1) + 1.

Original entry on oeis.org

5, 9, 33, 513, 131073, 8589934593, 36893488147419103233, 680564733841876926926749214863536422913, 231584178474632390847141970017375815706539969331281128078915168015826259279873

Offset: 0

Anonymous, May 17 2002

nonn

			a(0)=5 because 2^(2^0 + 1) + 1 = 2^(1 + 1) + 1 = 2^2 + 1 = 4 + 1 = 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..13

Cf. A000215, A006485, A077585.

[(2^((2^n)+1))+1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

2^(2^Range[0, 9] + 1) + 1 (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)

A070969(n):=(2^(2^n+1))+1$ makelist(A070969(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

a(n)=1<<(2^n+1)+1 \\ Charles R Greathouse IV, Jul 30 2011

Edited by Robert G. Wilson v, May 20 2002

A054868 Sum of bits of sum of bits of n: a(n) = wt(wt(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2

Offset: 0

Jeffrey Shallit, May 15 2000

			a(127) = 3 since 127 in base 2 is 1111111, whose sum of bits is 7 and 7 in base 2 is 111, whose sum of bits is 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340.
Michael Gilleland, Some Self-Similar Integer Sequences.
Index entries for sequences related to binary expansion of n.

Cf. A000120, A077585 (where records occur), A089224.

a054868 = a000120 . a000120  -- Reinhard Zumkeller, Mar 31 2015

a:= n-> (w-> w(w(n)))(k-> add(i, i=Bits[Split](k))):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 04 2022

a[n_] := DigitCount[DigitCount[n, 2, 1], 2, 1]; Array[a, 100, 0] (* Amiram Eldar, Jul 24 2023 *)

a(n) = norml2(binary(norml2(binary(n))))  \\ Michel Marcus, May 25 2013

a(n) = hammingweight(hammingweight(n)); \\ Ruud H.G. van Tol, Jul 03 2024

def a(n): return n.bit_count().bit_count()
print([a(n) for n in range(99)]) # Michael S. Branicky, Jul 04 2022

a(n) = A000120(A000120(n)).

a(2^(2^n-1)-1) = a(A077585(n)) = n (first occurrence). - Alois P. Heinz, Jul 04 2022

A135085 a(n) = A000110(2^n).

Original entry on oeis.org

1, 2, 15, 4140, 10480142147, 128064670049908713818925644, 172134143357358850934369963665272571125557575184049758045339873395

Offset: 0

Thomas Wieder, Nov 18 2007, Nov 19 2007

nonn

Number of set partitions of all subsets of a set, Bell(2^n).

			Let S={1,2,3,...,n} be a set of n elements and let SU be the set of all subsets of S including the empty set. The number of elements of SU is |SU| = 2^n. Now form all possible set partitions from SU including the empty set. This gives a set W and its number of elements is |W| = sum((stirling2(2^n,k)), k=0..2^n) = Bell(2^n).
For S={1,2} we have SU = { {}, {1}, {2}, {1,2} } and W =
{
{{{}}, {1}, {2}, {1, 2}},
{{2}, {1, 2}, {{}, {1}}},
{{1}, {1, 2}, {{}, {2}}},
{{1}, {2}, {{}, {1, 2}}},
{{{}}, {1, 2}, {{1}, {2}}},
{{{1}, {2}}, {{}, {1, 2}}},
{{1, 2}, {{}, {1}, {2}}},
{{{}}, {2}, {{1}, {1, 2}}},
{{{1}, {1, 2}}, {{}, {2}}},
{{2}, {{}, {1}, {1, 2}}},
{{{}}, {1}, {{2}, {1, 2}}},
{{{2}, {1, 2}}, {{}, {1}}},
{{1}, {{}, {2}, {1, 2}}},
{{{}}, {{1}, {2}, {1, 2}}},
{{{}, {1}, {2}, {1, 2}}}
}
and |W| = 15.

Cf. A000079, A000110, A008277, A077585, A135084.

ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n,nend,arg,k,w; nend:=5; for n from 0 to nend do arg:=2^n; w[n]:=sum((stirling2(arg,k)), k=0..arg); od; print(w[0],w[1],w[2],w[3],w[4],w[5],w[6],w[7],w[8],w[9],w[10]); end proc;

Table[BellB[2^n],{n,0,10}] (* Geoffrey Critzer, Jan 03 2014 *)

from sympy import bell
def A135085(n): return bell(2**n) # Chai Wah Wu, Jun 22 2022

a(n) = |W| = Sum_{k=0..2^n} Stirling2(2^n,k) = Bell(2^n), where Stirling2(n) is the Stirling number of the second kind and Bell(n) is the Bell number.

a(n) = exp(-1) * Sum_{k>=0} k^(2^n)/k!. - Ilya Gutkovskiy, Jun 13 2019

A135084 a(n) = A000110(2^n-1).

Original entry on oeis.org

1, 5, 877, 1382958545, 10293358946226376485095653, 8250771700405624889912456724304738028450190134337110943817172961

Offset: 1

Thomas Wieder, Nov 18 2007

nonn

Number of set partitions of all nonempty subsets of a set, Bell(2^n-1).

			Let S={1,2,3,...,n} be a set of n elements and let
SU be the set of all nonempty subsets of S. The number of elements of SU is |SU| = 2^n-1. Now form all possible set partitions from SU where the empty set is excluded. This gives a set W and its number of elements is |W| = Sum_{k=1..2^n-1} Stirling2(2^n-1,k).
For S={1,2} we have SU = { {1}, {2}, {1,2} } and W =
{
{{1}, {2}, {1, 2}},
{{1, 2}, {{1}, {2}}},
{{2}, {{1}, {1, 2}}},
{{1}, {{2}, {1, 2}}},
{{{1}, {2}, {1, 2}}}
}
and |W| = 5.

Amiram Eldar, Table of n, a(n) for n = 1..9

Cf. A000079, A000110, A008277, A077585, A135085.

ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n,nend,arg,k,w; nend:=5; for n from 1 to nend do arg:=2^n-1; w[n]:=sum((stirling2(arg,k)), k=1..arg); od; print(w[1],w[2],w[3],w[4],w[5],w[6],w[7],w[8],w[9],w[10]); end proc;

BellB[2^Range[6]-1] (* Harvey P. Dale, Jul 22 2012 *)

from sympy import bell
def A135084(n): return bell(2**n-1) # Chai Wah Wu, Jun 22 2022

a(n) = Sum_{k=1..2^n-1} Stirling2(2^n-1,k) = Bell(2^n-1), where Stirling2(n, k) is the Stirling number of the second kind and Bell(n) is the Bell number.

A263686 Smallest prime factor of double Mersenne numbers.

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 338193759479, 231733529, 62914441, 295257526626031

Offset: 1

Arkadiusz Wesolowski, Oct 23 2015

A double Mersenne number is a Mersenne number of the form 2^(2^p - 1) - 1, where p is a Mersenne exponent (A000043).
From M. F. Hasler, Feb 28 2025: (Start)
The prime factors of Mersenne numbers 2^q - 1 must be of the form 2*q*k + 1.
The four smallest double Mersenne numbers (p = 2, 3, 5, 7 => q = 3, 7, 31, 127) are prime, so their smallest prime factor is equal to themselves, a(n) = M(q). This is equivalent to k = (2^(q-1)-1)/q, which is almost as large as M(q) itself: k = 1, 9 and 34636833 for the first three terms, and for q = 127, k has just three digits less than M(q) = a(4) itself. The prime p = 11 is not a Mersenne exponent.
The fifth term, a(5) = 2*(2^13-1)*k + 1 with k = 20644229 (which is prime) is the first proper divisor of the respective M(q), as are the next three, corresponding to p = 17, 19 and 31.
For p = 61, M(q) has 694127911065419642 digits, and so far no factor is known, but it is known that it has no factor less than 10^36. (End)

Double Mersennes Prime Search, History
Wikipedia, Double Mersenne number

Cf. A000043, A000668, A001348, A020639, A049479, A077586, A122094. Subsequence of A016047. Subsequence of A309130.

forprime(p=2,,q=2^p-1; !ispseudoprime(q) && next(); if(ispseudoprime(2^q-1), print1(2^q-1,", ");next()); forstep(r=2*q+1,+oo,2*q, !ispseudoprime(r) && next(); if(Mod(2,r)^q-1 == 0, print1(r,", ");next(2)))) \\ Jeppe Stig Nielsen, Aug 28 2019

a(n) = spf(MM(A000043(n))) = A049479(A000668(n)), where spf = A020639 is the smallest prime factor, A049479 = spf o M, M(p) = 2^p-1 = A000225(p), MM = M o M = A077585, A000668(n) = M(A000043(n)), A000043 are the Mersenne prime exponents. - M. F. Hasler, Mar 01 2025

cp's OEIS Frontend

A000215 Fermat numbers: a(n) = 2^(2^n) + 1.

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A058891 a(n) = 2^(2^(n-1) - 1).

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A056220 a(n) = 2*n^2 - 1.

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A077586 a(n) = 2^(2^prime(n) - 1) - 1.

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A070969 a(n) = 2^(2^n + 1) + 1.

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A054868 Sum of bits of sum of bits of n: a(n) = wt(wt(n)).

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