

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
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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 n1 opened links (through n1 cuts), might be used to settle debt sequentially, with a golden link covering for unit cost. Here beside the n1 opened links, the n fragmented pieces have lengths n, 2*n, 4*n, ..., 2^(n1)*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 linkcost 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 20023 31(1) 3235.
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. 5051; 5758, University of Chicago Press, 1983.
D. Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, 17 (2014), #14.9.4.
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 entries for linear recurrences with constant coefficients, signature (5,8,4).


FORMULA

G.f.: x*(12*x+4*x^2) / ( (x1)*(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^n1, {n, 3*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)


PROG

(Haskell)
a003261 = (subtract 1) . a036289  Reinhard Zumkeller, Mar 05 2012
(PARI) A003261(n)=n*2^n1 \\ 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 A259214 A114246
Adjacent sequences: A003258 A003259 A003260 * A003262 A003263 A003264


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



