A014586 -id:A014586 - cp's OEIS frontend

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

Achim Flammenkamp

nonn

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)

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

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.

Cf. A025043, A014586-A014588, A355557.

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

A297963 The smallest position with nim-value n in subtract-a-square game.

Original entry on oeis.org

0, 1, 4, 25, 28, 29, 75, 103, 200, 234, 315, 364, 479, 633, 802, 1054, 1173, 1311, 1894, 2058, 2173, 2419, 3244, 3648, 4249, 4474, 4982, 5943, 6133, 6809, 7429, 8590, 8654, 9419, 10284, 11728, 12152, 13884, 15493, 16623, 17312, 18389, 19745, 20528, 22111, 23472

Offset: 0

David Eppstein, Jan 09 2018

nonn

a(n) is the position of the first copy of n in A014586.

			The sequence of nim-values for subtract-a-square (A014586) begins 0,1,0,1,2; the first position with value 2 is position 4 (starting from 0) so a(2)=4.

Nathan Fox, Subtraction Games with Infinite Subtraction Sets
Nathan Fox, Subtraction Games with Infinite Subtraction Sets [Local copy]

Cf. A014586.

A014587 Nim function for Take-a-Factorial-Game (a subtraction game).

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2

Offset: 0

Achim Flammenkamp

nonn

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

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

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.

Cf. A014586-A014589, A355556.

def A014587(max) :
    res = []
    fact = [1]
    while fact[-1] <= max : fact.append(factorial(len(fact)))
    for i in range(max+1) :
        moves = list({res[i-f] for f in fact if f <= i})
        moves.sort()
        k = len(moves)
        mex = next((j for j in range(k) if moves[j] != j), k)
        res.append(mex)
    return res
# Eric M. Schmidt, Jul 20 2013, corrected Eric M. Schmidt, Apr 24 2019

Conjecture: Appears to be periodic with period of length 25 =  [0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3] starting with the initial term (there is no preamble). - Michel Dekking, Jul 26 2019
This conjecture is false, because moving from 10! = 3628800 to 0 is a legal move, and so a(3628800) cannot be zero. A similar argument shows that for no value of P is this sequence periodic with period P starting at term 0 (for a(P!) cannot be zero). - Nathan Fox, Jul 28 2019.
The first counterexample to the conjecture above is a(5050) = 4. - Pontus von Brömssen, Jul 09 2022

A014588 Nim function for Take-a-Fibonacci-Game (a subtraction game).

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2

Offset: 0

Achim Flammenkamp

nonn

This game is also called Fibonacci nim, but there is also a different game with the same name. Its winning positions (the indexes of zeros in this sequence) are A001581 and its (much sparser) odd winning positions are A120904. - David Eppstein, Jun 14 2018

R. K. Guy, Unsolved Problems in Number Theory, E26.
David L. Silverman, Your Move, McGraw Hill, 1971, page 211. Reprinted by Dover Books, 1991 (mentions this game).

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.
Wikipedia, Sprague-Grundy theorem

Cf. A001581, A014586-A014589, A019307, A120904.

def A014588(max) :
    res = []
    for i in range(max+1) :
        moves = list({res[i-f] for f in fibonacci_xrange(1,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
# Eric M. Schmidt, Jul 20 2013, corrected Eric M. Schmidt, Apr 24 2019

cp's OEIS Frontend

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

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Sage

A297963 The smallest position with nim-value n in subtract-a-square game.

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A014587 Nim function for Take-a-Factorial-Game (a subtraction game).

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Sage

Formula

A014588 Nim function for Take-a-Fibonacci-Game (a subtraction game).

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Sage