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A003261 Woodall (or Riesel) numbers: n*2^n - 1.
(Formerly M4379)
26
1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519, 44040191, 92274687, 192937983, 402653183, 838860799, 1744830463, 3623878655, 7516192767 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

REFERENCES

A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 159.

K. R. Bhutani and A. B. Levin, "The Problem of Sawing a Chain", Journal of Recreational Mathematics 2002-3 31(1) 32-35.

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

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.

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=1..300

Ray Ballinger, Woodall Primes: Definition and Status

C. K. Caldwell, Woodall Numbers

Paul Leyland, Factors of Cullen and Woodall numbers

Paul Leyland, Generalized Cullen and Woodall numbers

Hisanori Mishima, Factorizations of many number sequences: Riesel numbers, n=1..100, n=101..200, n=201..300, n=301..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.

T. Sillke, Using Chains Links To Pay For A Room

Eric Weisstein's World of Mathematics, Woodall Number.

Wikipedia, Woodall number

Index to sequences with linear recurrences with constant coefficients, signature (5,-8,4).

FORMULA

G.f.: x*(-1-2*x+4*x^2) / ( (x-1)*(-1+2*x)^2 ). - Simon Plouffe in his 1992 dissertation.

Binomial transform of A133653 and double binomial transform of [1, 5, -1, 1, -1, 1,...]. - Gary W. Adamson, Sep 19 2007

MATHEMATICA

Table[n*2^n-1, {n, 3*4!}] [From Vladimir Joseph Stephan Orlovsky, Apr 25 2010]

PROG

(Haskell)

a003261 = (subtract 1) . a036289  -- Reinhard Zumkeller, Mar 05 2012

(PARI) A003261(n)=n*2^n-1  \\ - M. F. Hasler, Oct 31 2012

CROSSREFS

Cf. A002234, A002064, A005849, A050918.

a(n) = A036289(n)-1 = A002064(n)-2.

Cf. A133653.

Sequence in context: A077037 A201110 A220509 * A066187 A114246 A048457

Adjacent sequences:  A003258 A003259 A003260 * A003262 A003263 A003264

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 23 11:24 EDT 2014. Contains 245993 sequences.