

A006022


SpragueGrundy (or Nim) values for the game of Maundy cake on an n X 1 sheet.
(Formerly M2219)


4



0, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 10, 1, 8, 6, 15, 1, 13, 1, 16, 8, 12, 1, 22, 6, 14, 13, 22, 1, 21, 1, 31, 12, 18, 8, 31, 1, 20, 14, 36, 1, 29, 1, 34, 21, 24, 1, 46, 8, 31, 18, 40, 1, 40, 12, 50, 20, 30, 1, 51, 1, 32, 29, 63, 14, 45, 1, 52, 24, 43, 1, 67, 1, 38, 31, 58, 12, 53, 1
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OFFSET

1,4


COMMENTS

There are three equivalent formulas for a(n). Suppose n >= 2, and let p1 <= p2 <= ... <= pk be the prime factors of n, with repetition.
Theorem 1: a(1) = 0. For n >= 2, a(n) = n*s(n), where
s(n) = 1/p1 + 1/(p1*p2) + 1/(p1*p2*p3) + ... + 1/(p1*p2*...*pk).
This is implicit in Berlekamp, Conway and Guy, Winning Ways, 2 vols., 1982, pp. 28, 53.
Note that s(n) = A322034(n) / A322035(n).
_David Sycamore_ observed on Nov 24 2018 that Theorem 1 implies a(n) < n for all n (see comments in A322034), and also leads to a simple recurrence for a(n):
Theorem 2: a(1) = 0. For n >= 2, a(n) = p*a(n/p) + 1, where p is the largest prime factor of n.
Proof. (Th. 1 implies Th. 2) If n is a prime, Theorem 1 gives a(n) = 1 = n*a(1)+1. For a nonprime n, let n = m*p where p is the largest prime factor of n and m >= 2. From Theorem 1, a(m) = m*s(m), a(n) = q*m*(s(m) + 1/n) = q*a(m) + 1.
(Th. 2 implies Th. 1) The reverse implication is equally easy.
Theorem 2 is equivalent to the following more complicated recurrence:
Theorem 3: a(1) = 1. For n >= 2, a(n) = max_{pn, p prime} (p*a(n/p)+1).


REFERENCES

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 28, 53.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Second Edition, Vol. 1, A K Peters, 2001, pp. 27, 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000


EXAMPLE

For n=24, s(24) = 1/2 + 1/4 + 1/8 + 1/24 = 11/12, so a(24) = 24*11/12 = 22.


MAPLE

# from N. J. A. Sloane, Nov 28 2018
P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: # A027746
s:=proc(n) local i, t, b; global P; t:=0; b:=1; for i in [P(n)] do b:=b*i; t:=t+1/b; od; t; end; # A322034/A0322035
A006022 := n > if n = 1 then 0 else n*s(n); fi;


MATHEMATICA

Nest[Function[{a, n}, Append[a, Max@ Map[# a[[n/#]] + 1 &, Rest@ Divisors@ n]]] @@ {#, Length@ # + 1} &, {0, 1}, 77] (* Michael De Vlieger, Nov 23 2018 *)


PROG

(Haskell)
a006022 1 = 0
a006022 n = (+ 1) $ sum $ takeWhile (> 1) $
iterate (\x > x `div` a020639 x) (a032742 n)
 Reinhard Zumkeller, Jun 03 2012
(PARI) lista(nn) = {my(v = vector(nn)); for (n=1, nn, if (n>1, my(m = 0); fordiv (n, d, if (d>1, m = max(m, d*v[n/d]+1))); v[n] = m; ); print1(v[n], ", "); ); } \\ Michel Marcus, Nov 25 2018


CROSSREFS

Cf. A020639, A032742, A001348, A002182, A008864.
Cf. also A027746, A322034, A322035, A322036.
Sequence in context: A295276 A301856 A301829 * A078896 A322582 A273133
Adjacent sequences: A006019 A006020 A006021 * A006023 A006024 A006025


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

Edited and extended by Christian G. Bower, Oct 18 2002
Entry revised by N. J. A. Sloane, Nov 28 2018


STATUS

approved



