A002064 Cullen numbers: a(n) = n*2^n + 1.

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, 66571993089

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

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, 5 + 4 = 9 = a(2); for 5/4 + 7/8 = 17/8, 17 + 8 = 25 = a(3); for 17/8 + 15/16 = 49/16, 49 + 16 = 65 = a(4); for 49/16 + 31/32 = 129/32, 129 + 32 = 161 = a(5). For each pairwise sum a/b, a + b = n*2^(n+1). - J. M. Bergot, May 06 2015

Number of divisors of (2^n)^(2^n). - Gus Wiseman, May 03 2021

Named after the Irish Jesuit priest James Cullen (1867-1933), who checked the primality of the terms up to n=100. - Amiram Eldar, Jun 05 2021

Examples

			G.f. = 1 + 3*x + 9*x^2 + 25*x^3 + 65*x^4 + 161*x^5 + 385*x^6 + 897*x^7 + ... - _Michael Somos_, Jul 18 2018
		

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.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 240-242.
• 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.
• Attila Bérczes, István Pink, and Paul Thomas Young, Cullen numbers and Woodall numbers in generalized Fibonacci sequences, J. Num. Theor. (2024) Vol. 262, 86-102.
• Yuri Bilu, Diego Marques, and Alain Togbé, Generalized Cullen numbers in linear recurrence sequences, Journal of Number Theory, Vol. 202 (2019), pp. 412-425; arXiv preprint, arXiv:1806.09441 [math.NT], 2018.
• Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
• C. K. Caldwell, The Top Twenty: Cullen Primes.
• James Cullen, Question 15897, Educational Times, Vol. 58 (December 1905), p. 534.
• Orhan Eren and Yüksel Soykan, Gaussian Generalized Woodall Numbers, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
• Orhan Eren and Yüksel Soykan, On Dual Hyperbolic Generalized Woodall Numbers, Archives Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
• Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
• José María Grau and Florian Luca, Cullen numbers with the Lehmer property, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134; arXiv preprint, arXiv:1103.3578 [math.NT], Mar 18 2011.
• Paul Leyland, Factors of Cullen and Woodall numbers.
• Paul Leyland, Generalized Cullen and Woodall numbers.
• Diego Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 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, Appendix to Thesis, Montreal, 1992.
• Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
• Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
• Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences, Vol. 8, No. 4 (2019), pp. 87-92.
• Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
• Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences, Vol. 8, No. 10 (2019).
• Eric Weisstein's World of Mathematics, Cullen Number.
• Wikipedia, Cullen number.
• Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Crossrefs

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

Cf. A000325, A186947.

Diagonal k = n + 1 of A046688.

A000005 counts divisors of n.

A000312 = n^n.

A002109 gives hyperfactorials (sigma: A260146, omega: A303281).

A057156 = (2^n)^(2^n).

A062319 counts divisors of n^n.

A173339 lists positions of squares in A062319.

A188385 gives the highest prime exponent in n^n.

A249784 counts divisors of n^n^n.

Cf. A000169, A000272, A036289, A066959, A176029, A340841, A343656.

Programs

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

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 *)

PARI

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

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

a(n) = 2^n * A000325(n) = 4^n * A186947(-n) for all n in Z. - Michael Somos, Jul 18 2018

a(n) = Sum_{i=0..n-1} a(i) + A000325(n+1). - Ivan N. Ianakiev, Aug 07 2019

a(n) = sigma((2^n)^(2^n)) = A000005(A057156(n)) = A062319(2^n). - Gus Wiseman, May 03 2021

Sum_{n>=0} 1/a(n) = A340841. - Amiram Eldar, Jun 05 2021

Extensions

Edited by M. F. Hasler, Oct 31 2012