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 A006022 Sprague-Grundy (or Nim) values for the game of Maundy cake on an n X 1 sheet. (Formerly M2219) 7
 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 James 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 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; # N. J. A. Sloane, Nov 28 2018 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 * A326065 A078896 A322582 Adjacent sequences:  A006019 A006020 A006021 * A006023 A006024 A006025 KEYWORD nonn AUTHOR 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 October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)