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A019567 a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible by 4n + 1. 2
1, 2, 3, 6, 4, 6, 10, 14, 5, 18, 10, 12, 21, 26, 9, 30, 6, 22, 9, 30, 27, 8, 11, 10, 24, 50, 12, 18, 14, 12, 55, 50, 7, 18, 34, 46, 14, 74, 24, 26, 33, 20, 78, 86, 29, 90, 18, 18, 48, 98, 33, 10, 45, 70, 15, 24, 60, 38, 29, 78, 12, 84, 41, 110, 8, 84, 26, 134, 12, 46, 35, 36, 68, 146 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Write down 1, then 2 to left, 3 to right, 4 to left, ..., getting [ 2n,2n-2,...,4,2,1,3,5,...,2n-1 ]; the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives order of permutation sending 1 to 2n, 2 to 2n-2, ..., 2n to 2n-1.

Equivalently, the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives the number of Mongean shuffles needed to return a deck of 2n cards (n=1,2,3,...) to its original order.

REFERENCES

A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; see pp. 134-135.

W. W. Rouse Ball, Mathematical Recreations and Essays, 11th ed. 1939, p. 311

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..2000

P. Diaconis, The mathematics of perfect shuffles, Adv. Appl. Math. 4 (2) (1983) 175-196.

E. Ross, Methematics and Music: The Mathieu Group M_12 (2011), Chapter 2.

EXAMPLE

Illustrating the initial terms:

n 4n+1 2^m+1 2^m-1 m

0..1...........1...1

1..5.....5.........2

2..9.....9.........3

3.13...5*13........6

4.17.....17........4

5.21..........3*21.6

6.25..41*25.......10

MAPLE

A019567:=  proc(n)

    for m from 1 do

        if modp(2^m-1, 4*n+1) =0 or modp(2^m+1, 4*n+1)=0 then

            return m ;

        end if;

    end do;

end proc: # N. J. A. Sloane, Jul 28 2007

CROSSREFS

Sequence in context: A209775 A125703 A156688 * A098286 A226615 A138608

Adjacent sequences:  A019564 A019565 A019566 * A019568 A019569 A019570

KEYWORD

nonn,easy

AUTHOR

John Bullitt (metta(AT)world.std.com), N. J. A. Sloane and J. H. Conway

EXTENSIONS

Comments corrected by Mikko Nieminen, Jul 26 2007, who also provided the Domoryad reference.

STATUS

approved

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Last modified November 23 00:51 EST 2014. Contains 249836 sequences.