Game values of arithmetic functions. (English) Zbl 1490.91038
Summary: Arithmetic functions in number theory meet the Sprague-Grundy function from combinatorial game theory. We study a variety of 2-player games induced by standard arithmetic functions, such as Euclidian division, divisors, remainders and relatively prime numbers, and their negations.
MSC:
| 91A46 | Combinatorial games |
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 91A05 | 2-person games |
Software:
OEISOnline Encyclopedia of Integer Sequences:
Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).The ruler function: exponent of the highest power of 2 dividing 2n. Equivalently, the 2-adic valuation of 2n.
Kimberling’s paraphrases: if n = (2k-1)*2^m then a(n) = k.
Smallest prime dividing n is a(n)-th prime (a(1)=0).
References:
| [1] | E. R. Berlekamp, J. H. Conway, R. K. Guy. Winning Ways, Academic Press, London, 1982. · Zbl 0485.00025 |
| [2] | C. Bouton, Nim, a game with a complete mathematical theory, Annals of Math., 2nd ser. 3 (1901-2), 35-39. · JFM 32.0225.02 |
| [3] | A. Dailly, E. Duchene, U. Larsson, G. Paris, Partition games, Discrete Appl. Math. 285 (2020) 509-525. · Zbl 1452.91056 |
| [4] | P. Grundy, Mathematics and games, Eureka 2 (1939), 6-8. |
| [5] | G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (5th ed.), The Clarendon Press, Oxford University Press, New York, 1979. · Zbl 0423.10001 |
| [6] | N. Mc Kay, U. Larsson, R. J. Nowakowski, A. Siegel, Wythoff partizan subtraction, Int. J. Game Theory 47, Special Issue on Combinatorial Games, 2018. Invited paper from Combina-torial Game Theory Colloquium, Lisbon 2015 (2018) 613-652. · Zbl 1457.91115 |
| [7] | H. Shapiro, An Arithmetic Function Arising from the Φ Function, Am. Math. Mon. 50:1 (1943), 18-30. · Zbl 0061.08002 |
| [8] | N. Sloane, The On-Line Encyclopedia of Integer Sequences (OEIS), website at http://oeis.org/. · Zbl 1044.11108 |
| [9] | R. Sprague,Über mathematische Kampfspiele, Tôhoku J. Math. 41 (1936), 438-444. · JFM 62.1070.03 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
