A014589 Nim function for Take-a-Prime (or Subtract-a-Prime) Game.

0, 0, 1, 1, 2, 2, 3, 3, 4, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 0, 4, 1, 5, 2, 6, 3, 4, 7, 0, 0, 1, 1, 2, 2, 3, 3, 4, 8, 5, 7, 6, 8, 9, 0, 4, 1, 5, 2, 6, 0, 4, 1, 5, 2, 6, 3, 4, 7, 5, 8, 4, 10, 5, 7, 6, 8, 4, 7, 5, 8, 6, 10, 9, 7, 4, 8, 5, 10, 6, 0, 4, 1, 5, 2, 6, 0, 4, 1, 5, 2, 6, 3

Offset: 0

Author

Achim Flammenkamp

Keywords

• nonn

Comments

The zero positions are given by A025043. - Nathan Fox, May 21 2013

Concerning the January 1997 dissertation of Achim Flammenkamp, his home page (currently http://wwwhomes.uni-bielefeld.de/cgi-bin/cgiwrap/achim/index.cgi) has the link shown below, and a comment that a book was published in July 1997 by Hans-Jacobs-Verlag, Lage, Germany with the title Lange Perioden in Subtraktions-Spielen (ISBN 3-932136-10-1). This is an enlarged study (more than 200 pages) of his dissertation. - N. J. A. Sloane, Jul 25 2019

As noted by Alexis Huet, a(n) <= 11 for all n <= 32452842 (see links). - Pontus von Brömssen, Jul 09 2022

From Bert Dobbelaere, Apr 09 2024: (Start)

For n <= 10^9, a(n) <= 11.

For even n <= 10^9, if a(n)=0, n is in {0, 10, 34, 100, 310}.

For even n <= 10^9, if a(n)=1, n is in {2, 12, 36, 102, 312}.

For even n <= 10^9, if a(n)=2, n is in {4, 14, 38, 104, 314, 1574}.

For even n <= 10^9, if a(n)=3, n is in {6, 16, 40, 106, 316, 1576, 1996, 5566}.

The only odd n <= 10^9 for which a(n)=4 is 17.

The only odd n <= 10^9 for which a(n)=5 is 19.

The only odd n <= 10^9 for which a(n)=6 is 21.

The only even n <= 10^9 for which a(n)=7 is 24.

There are no even n <= 10^9 for which a(n)=8 or a(n)=10.

There are no odd n <= 10^9 for which a(n)=11. (End)

References

• R. K. Guy, Unsolved Problems in Number Theory, E26.

Links

• Eric M. Schmidt, Table of n, a(n) for n = 0..10000
• Achim Flammenkamp, Lange Perioden in Subtraktions-Spielen, Dissertation, Dept. Math., University of Bielefeld, Germany.
• Alexis Huet, First 32452843 terms.
• Alexis Huet, Nim function for take-a-prime game.

Crossrefs

Cf. A025043, A014586-A014588, A355557.

Programs

Sage

def A014589_list(max) :
    res = []
    for i in range(max+1) :
        moves = list({res[i-p] for p in prime_range(i+1)})
        moves.sort()
        k = len(moves)
        mex = next((j for j in range(k) if moves[j] != j), k)
        res.append(mex)
    return res
print(A014589_list(50))
# Eric M. Schmidt, Jul 20 2013, corrected Eric M. Schmidt, Apr 24 2019