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A006022 Sprague-Grundy (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 (list; graph; refs; listen; history; text; internal format)
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_{p|n, 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

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Last modified January 22 06:25 EST 2019. Contains 319353 sequences. (Running on oeis4.)