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A002064 Cullen numbers: n*2^n + 1.
(Formerly M2795 N1125)
45
1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry, Jun 12 2003

Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1] =-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010

A composite integer m is called a Lehmer number, or has the Lehmer property if phi(m) | m - 1, where phi(m) is the Euler function of m = A000010(m). According to the Grau and Luca reference's abstract they "show that there is no Cullen number with [the Lehmer property]". Hence, if phi (C_n) | C_n - 1, then C_n is prime. - Jonathan Vos Post, Mar 20 2011

Partial sums of this sequence A002064 are listed in A181527. - Vladimir Joseph Stephan Orlovsky, Jul 09 2011.

Indices of primes are listed in A005849. - M. F. Hasler, Jan 18 2015

Add the list of fractions beginning with 1/2 + 3/4 + 7/8 + ....2^n-1/2^n and take the sums pairwise from left to right. For 1/2 + 3/4 = 5/4 with 5+4=9=a(2); for 5/4 + 7/8 = 17/8 with 17+8=25=a(3); for 17/8 + 15/16= 49/16 with 49+16=65=a(4); 49/16 +31/32=129/32 with 129+32=161=a(5). For each pairwise sum a/b, a+b=n*2^(n+1). - J. M. Bergot, May 06 2015

REFERENCES

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

R. K. Guy, Unsolved Problems in Number Theory, B20.

W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 346.

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..300

Ray Ballinger, Cullen Primes: Definition and Status

C. K. Caldwell, The Top Twenty: Cullen Primes

Jose Maria Grau, Florian Luca, Cullen Numbers with the Lehmer Property, arXiv:1103.3578 [math.NT],  Mar 18 2011.

Paul Leyland, Factors of Cullen and Woodall numbers

Paul Leyland, Generalized Cullen and Woodall numbers

D. Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, 17 (2014), #14.9.4.

Hisanori Mishima, Factorizations of many number sequences, Cullen numbers (n = 1 to 100), (n = 101 to 200), (n = 201 to 300), (n = 301 to 323)

Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.

Eric Weisstein's World of Mathematics, Cullen Number.

Wikipedia, Cullen number

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

FORMULA

a(n) = 4a(n-1) - 4a(n-2) + 1. - Paul Barry, Jun 12 2003

a(n) = sum of row (n+1) of triangle A130197. Example: a(3) = 25 = (12 + 8 + 4 + 1), row 4 of A130197. - Gary W. Adamson, May 16 2007

Row sums of triangle A134081. - Gary W. Adamson, Oct 07 2007

Equals row sums of triangle A143038. - Gary W. Adamson, Jul 18 2008

Equals row sums of triangle A156708. - Gary W. Adamson, Feb 13 2009

G.f.: -(1-2*x+2*x^2)/((-1+x)*(2*x-1)^2). a(n) = A001787(n+1)+1-A000079(n). - R. J. Mathar, Nov 16 2007

a(n) = 1 + 2^(n + log_2(n)) ~ 1 + A000079(n+A004257(n)). a(n) ~ A000051(n+A004257(n)). - Jonathan Vos Post, Jul 20 2008

a(0)=1, a(1)=3, a(2)=9, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale, Oct 13 2011

a(n) = A036289(n) + 1 = A003261(n) + 2. - Reinhard Zumkeller, Mar 16 2013

E.g.f.: 2*x*exp(2*x) + exp(x). - Robert Israel, Dec 12 2014

MAPLE

A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

Table[n*2^n+1, {n, 0, 2*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)

LinearRecurrence[{5, -8, 4}, {1, 3, 9}, 51] (* Harvey P. Dale, Oct 13 2011 *)

CoefficientList[Series[(1 - 2 x + 2 x^2)/((1 - x) (2 x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)

PROG

(PARI) A002064(n)=n*2^n+1  \\ M. F. Hasler, Oct 31 2012

(Haskell)

a002064 n = n * 2 ^ n + 1

a002064_list = 1 : 3 : (map (+ 1) $ zipWith (-) (tail xs) xs)

   where xs = map (* 4) a002064_list

-- Reinhard Zumkeller, Mar 16 2013

(MAGMA) [n*2^n + 1: n in [0..40]]; // Vincenzo Librandi, May 07 2015

CROSSREFS

Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.

Sequence in context: A145127 A096260 A195417 * A129589 A096322 A058396

Adjacent sequences:  A002061 A002062 A002063 * A002065 A002066 A002067

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by M. F. Hasler, Oct 31 2012

STATUS

approved

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Last modified July 25 04:11 EDT 2016. Contains 275025 sequences.