cp's OEIS Frontend

A003261 Woodall (or Riesel) numbers: n*2^n - 1.

%I A003261 M4379 #145 Aug 05 2025 10:41:37
%S A003261 1,7,23,63,159,383,895,2047,4607,10239,22527,49151,106495,229375,
%T A003261 491519,1048575,2228223,4718591,9961471,20971519,44040191,92274687,
%U A003261 192937983,402653183,838860799,1744830463,3623878655,7516192767,15569256447,32212254719,66571993087
%N A003261 Woodall (or Riesel) numbers: n*2^n - 1.
%C A003261 For n>1, a(n) is base at which zero is reached for the function "write f(j) in base j, read as base j+1 and then subtract 1 to give f(j+1)" starting from f(n) = n^2 - 1. - _Henry Bottomley_, Aug 06 2000
%C A003261 Sequence corresponds also to the maximum chain length of the classic puzzle whereby, under agreed commercial terms, an asset of unringed golden chain, when judiciously fragmented into as few as n pieces and n-1 opened links (through n-1 cuts), might be used to settle debt sequentially, with a golden link covering for unit cost. Here beside the n-1 opened links, the n fragmented pieces have lengths n, 2*n, 4*n, ..., 2^(n-1)*n. For instance, the chain of original length a(5)=159, if segregated by 4 cuts into 5+1+10+1+20+1+40+1+80, may be used to pay sequentially, i.e., a link-cost at a time, for an equivalent cost up to 159 links, to the same creditor. - _Lekraj Beedassy_, Feb 06 2003
%D A003261 A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 159.
%D A003261 K. R. Bhutani and A. B. Levin, "The Problem of Sawing a Chain", Journal of Recreational Mathematics 2002-3 31(1) 32-35.
%D A003261 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A003261 M. Gardner, Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American, "Gold Links", Problem 4, pp. 50-51; 57-58, University of Chicago Press, 1983.
%D A003261 O. O'Shea, Mathematical Brainteasers with Surprising Solutions, Problem 76, pp. 183-185, Prometheus Books, Guilford, Connecticut, 2020.
%D A003261 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 241.
%D A003261 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003261 Vladimir Pletser, <a href="/A003261/b003261.txt">Table of n, a(n) for n = 1..3000</a> (terms 1..300 from T. D. Noe).
%H A003261 Ray Ballinger, <a href="http://web.archive.org/web/20161028080439/http://www.prothsearch.net/woodall.html">Woodall Primes: Definition and Status</a>.
%H A003261 Attila Bérczes, István Pink, and Paul Thomas Young, <a href="https://doi.org/10.1016/j.jnt.2024.03.006">Cullen numbers and Woodall numbers in generalized Fibonacci sequences</a>, J. Num. Theor. (2024) Vol. 262, 86-102.
%H A003261 Alfred Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972. See p. 159.
%H A003261 C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=WoodallNumber">Woodall Numbers</a>.
%H A003261 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 A003261 Orhan Eren and Yüksel Soykan, <a href="https://doi.org/10.9734/acri/2024/v24i11981">On Dual Hyperbolic Generalized Woodall Numbers</a>, Arch. Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
%H A003261 Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, <a href="https://arxiv.org/abs/2507.23619">Picturesque convolution-like recurrences and partial sums' generation</a>, arXiv:2507.23619 [math.NT], 2025. See p. 27.
%H A003261 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/cw.htm">Factors of Cullen and Woodall numbers</a>.
%H A003261 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/gcw.htm">Generalized Cullen and Woodall numbers</a>.
%H A003261 D. 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, 17 (2014), #14.9.4.
%H A003261 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha122.htm">Factorizations of many number sequences: Riesel numbers, n=1..100</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha123.htm">n=101..200</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha124.htm">n=201..300</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha125.htm">n=301..323</a>.
%H A003261 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; arXiv:0911.4975 [math.NT], 2009.
%H A003261 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A003261 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/chain-link-pay">Using Chains Links To Pay For A Room</a>.
%H A003261 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 A003261 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 (2019) Vol. 8, No. 4, 87-92.
%H A003261 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 A003261 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 (2019) Vol. 8, No. 10.
%H A003261 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WoodallNumber.html">Woodall Number</a>.
%H A003261 Wikipedia, <a href="http://en.wikipedia.org/wiki/Woodall_number">Woodall number</a>.
%H A003261 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,4).
%F A003261 G.f.: x*(-1-2*x+4*x^2) / ( (x-1)*(-1+2*x)^2 ). - _Simon Plouffe_ in his 1992 dissertation
%F A003261 Binomial transform of A133653 and double binomial transform of [1, 5, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Sep 19 2007
%F A003261 a(n) = -(2)^n * A006127(-n) for all n in Z. - _Michael Somos_, Nov 04 2018
%F A003261 E.g.f.: 1 + exp(x)*(2*exp(x)*x - 1). - _Stefano Spezia_, Nov 24 2024
%e A003261 G.f. = x + 7*x^2 + 23*x^3 + 63*x^4 + 159*x^5 + 383*x^6 + 895*x^7 + ... - _Michael Somos_, Nov 04 2018
%p A003261 for n from 1 to 3000 do n, n*2^n -1; end do; # _Vladimir Pletser_, Dec 30 2022
%t A003261 Table[n*2^n-1,{n,3*4!}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 25 2010 *)
%t A003261 LinearRecurrence[{5,-8,4},{1,7,23},30] (* _Harvey P. Dale_, Mar 13 2022 *)
%o A003261 (Haskell)
%o A003261 a003261 = (subtract 1) . a036289  -- _Reinhard Zumkeller_, Mar 05 2012
%o A003261 (PARI) A003261(n)=n*2^n-1  \\ _M. F. Hasler_, Oct 31 2012
%o A003261 (Magma) [n*2^n - 1: n in [1..30]]; // _G. C. Greubel_, Nov 04 2018
%o A003261 (Python) [n*2**n - 1 for n in range(1, 29)] # _Michael S. Branicky_, Jan 07 2021
%Y A003261 Cf. A002234, A002064, A005849, A050918, A006127.
%Y A003261 a(n) = A036289(n) - 1 = A002064(n) - 2.
%Y A003261 Cf. A133653.
%K A003261 nonn,easy,nice
%O A003261 1,2
%A A003261 _N. J. A. Sloane_