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%I M2795 N1125 #174 Jan 25 2024 11:00:50
%S 1,3,9,25,65,161,385,897,2049,4609,10241,22529,49153,106497,229377,
%T 491521,1048577,2228225,4718593,9961473,20971521,44040193,92274689,
%U 192937985,402653185,838860801,1744830465,3623878657,7516192769,15569256449,32212254721
%N Cullen numbers: a(n) = n*2^n + 1.
%C Binomial transform is A084859. Inverse binomial transform is A004277. - _Paul Barry_, Jun 12 2003
%C 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
%C Indices of primes are listed in A005849. - _M. F. Hasler_, Jan 18 2015
%C 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
%C Number of divisors of (2^n)^(2^n). - _Gus Wiseman_, May 03 2021
%C 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
%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D R. K. Guy, Unsolved Problems in Number Theory, B20.
%D W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 346.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002064/b002064.txt">Table of n, a(n) for n=0..300</a>
%H Ray Ballinger, <a href="http://web.archive.org/web/20161028015144/http://www.prothsearch.net/cullen.html">Cullen Primes: Definition and Status</a>.
%H Yuri Bilu, Diego Marques, and Alain Togbé, <a href="https://doi.org/10.1016/j.jnt.2018.11.025">Generalized Cullen numbers in linear recurrence sequences</a>, Journal of Number Theory, Vol. 202 (2019), pp. 412-425; <a href="https://arxiv.org/abs/1806.09441">arXiv preprint</a>, arXiv:1806.09441 [math.NT], 2018.
%H Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, <a href="https://arxiv.org/abs/2108.04302">Restricted generating trees for weak orderings</a>, arXiv:2108.04302 [math.CO], 2021.
%H C. K. Caldwell, <a href="https://t5k.org/top20/page.php?id=6">The Top Twenty: Cullen Primes</a>.
%H James Cullen, <a href="https://archive.org/details/educationaltimes58educ/page/534/mode/2up">Question 15897</a>, Educational Times, Vol. 58 (December 1905), p. 534.
%H Orhan Eren and Yüksel Soykan, <a href="https://doi.org/10.9734/ACRI/2023/v23i8611">Gaussian Generalized Woodall Numbers</a>, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
%H Jon Grantham and Hester Graves, <a href="https://arxiv.org/abs/2009.04052">The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits</a>, arXiv:2009.04052 [math.NT], 2020.
%H José María Grau and Florian Luca, <a href="https://doi.org/10.1090/S0002-9939-2011-10899-2">Cullen numbers with the Lehmer property</a>, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134; <a href="http://arxiv.org/abs/1103.3578">arXiv preprint</a>, arXiv:1103.3578 [math.NT], Mar 18 2011.
%H Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/cw.htm">Factors of Cullen and Woodall numbers</a>.
%H Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/gcw.htm">Generalized Cullen and Woodall numbers</a>.
%H Diego Marques, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Marques/marques5r2.html">On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.4.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/">Factorizations of many number sequences</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha118.htm">Cullen numbers (n = 1 to 100)</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha119.htm">(n = 101 to 200)</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha120.htm">(n = 201 to 300)</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha121.htm">(n = 301 to 323)</a>.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de Séries Génératrices et Quelques Conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Wacław Sierpiński, <a href="http://matwbn.icm.edu.pl/kstresc.php?tom=42&wyd=10">Elementary Theory of Numbers</a>, Warszawa 1964.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.2634312">On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers</a>, Politecnico di Torino (Italy, 2019).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2044">Composition Operations of Generalized Entropies Applied to the Study of Numbers</a>, International Journal of Sciences, Vol. 8, No. 4 (2019), pp. 87-92.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences, Vol. 8, No. 10 (2019).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CullenNumber.html">Cullen Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cullen_prime">Cullen number</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,4).
%F a(n) = 4a(n-1) - 4a(n-2) + 1. - _Paul Barry_, Jun 12 2003
%F 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
%F Row sums of triangle A134081. - _Gary W. Adamson_, Oct 07 2007
%F Equals row sums of triangle A143038. - _Gary W. Adamson_, Jul 18 2008
%F Equals row sums of triangle A156708. - _Gary W. Adamson_, Feb 13 2009
%F 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
%F 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
%F 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
%F a(n) = A036289(n) + 1 = A003261(n) + 2. - _Reinhard Zumkeller_, Mar 16 2013
%F E.g.f.: 2*x*exp(2*x) + exp(x). - _Robert Israel_, Dec 12 2014
%F a(n) = 2^n * A000325(n) = 4^n * A186947(-n) for all n in Z. - _Michael Somos_, Jul 18 2018
%F a(n) = Sum_{i=0..n-1} a(i) + A000325(n+1). - _Ivan N. Ianakiev_, Aug 07 2019
%F a(n) = sigma((2^n)^(2^n)) = A000005(A057156(n)) = A062319(2^n). - _Gus Wiseman_, May 03 2021
%F Sum_{n>=0} 1/a(n) = A340841. - _Amiram Eldar_, Jun 05 2021
%e 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
%p A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[n*2^n+1,{n,0,2*4!}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 25 2010 *)
%t LinearRecurrence[{5,-8,4},{1,3,9},51] (* _Harvey P. Dale_, Oct 13 2011 *)
%t CoefficientList[Series[(1 - 2 x + 2 x^2)/((1 - x) (2 x - 1)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 07 2015 *)
%o (PARI) A002064(n)=n*2^n+1 \\ _M. F. Hasler_, Oct 31 2012
%o (Haskell)
%o a002064 n = n * 2 ^ n + 1
%o a002064_list = 1 : 3 : (map (+ 1) $ zipWith (-) (tail xs) xs)
%o where xs = map (* 4) a002064_list
%o -- _Reinhard Zumkeller_, Mar 16 2013
%o (Magma) [n*2^n + 1: n in [0..40]]; // _Vincenzo Librandi_, May 07 2015
%Y Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.
%Y Cf. A000325, A186947.
%Y Diagonal k = n + 1 of A046688.
%Y A000005 counts divisors of n.
%Y A000312 = n^n.
%Y A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
%Y A057156 = (2^n)^(2^n).
%Y A062319 counts divisors of n^n.
%Y A173339 lists positions of squares in A062319.
%Y A188385 gives the highest prime exponent in n^n.
%Y A249784 counts divisors of n^n^n.
%Y Cf. A000169, A000272, A036289, A066959, A176029, A340841, A343656.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _M. F. Hasler_, Oct 31 2012
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