A306264 -id:A306264 - cp's OEIS frontend

A006022 Sprague-Grundy (or Nim) values for the game of Maundy cake on an n X 1 sheet.

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

Offset: 1

N. J. A. Sloane

nonn

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) = 0. For n >= 2, a(n) = max_{p|n, p prime} (p*a(n/p)+1).

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

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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jonathan Blanchette and Robert Laganière, A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting, arXiv:1910.11749 [cs.DS], 2019.

Cf. A020639, A032742, A001348, A002182, A008864.
Cf. also A027746, A306264, A306363, A322034, A322035, A322036, A322382, A328898, A333783, A332993, A333791.
Row sums of A328969.

a006022 1 = 0
a006022 n = (+ 1) $ sum $ takeWhile (> 1) $
          iterate (\x -> x `div` a020639 x) (a032742 n)
-- Reinhard Zumkeller, Jun 03 2012

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/A322035
A006022 := n -> if n = 1 then 0 else n*s(n); fi;
# N. J. A. Sloane, Nov 28 2018

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 *)

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

a(n) = n * Sum_{k=1..N} (1/(p1^m1*p2^m2*...*pk^mk)) * (pk^mk-1)/(pk-1) for n>=2, where pk is the k-th distinct prime factor of n, N is the number of distinct prime factors of n, and mk is the multiplicity of pk occurring in n. To prove this, expand the factors in Theorem 1 and use the geometrical series identity. - Jonathan Blanchette, Nov 01 2019
From Antti Karttunen, Apr 12 2020: (Start)
a(n) = A322382(n) + A333791(n).
a(n) = A332993(n) - n = A001065(n) - A333783(n). (End)
a(n) = Sum_{k=1..bigomega(n)} F^k(n), where F^k(n) is the k-th iterate of F(n) = A032742(n). - Ridouane Oudra, Jan 26 2024

Edited and extended by Christian G. Bower, Oct 18 2002

Entry revised by N. J. A. Sloane, Nov 28 2018

A324388 If n is a prime power (in A000961), then a(n) = n, otherwise a(n) is the greatest proper unitary divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 15, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 21, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 20, 61, 31, 9, 64, 13, 33, 67, 17, 23, 35, 71, 9, 73, 37, 25, 19, 11, 39, 79, 16, 81, 41, 83, 28, 17, 43, 29, 11, 89, 45

Offset: 1

Antti Karttunen and David James Sycamore, Feb 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, with a color function showing prime n in red, n that is a proper prime power in gold, n that is composite and squarefree in green, and n that is neither squarefree nor prime power in blue and magenta, where magenta signifies powerful n that is not a prime power.

Cf. A000961, A306264.

a[n_] := If[PrimePowerQ[n], n, SelectFirst[Transpose@ {Reverse@ #[[-Ceiling[Length[#]/2] ;; -2]], #[[2 ;; Ceiling[Length[#]/2]]]} &@ Divisors[n], CoprimeQ @@ # &][[1]] ]; a[1] = 1; Array[a, 120] (* Michael De Vlieger, Jun 24 2025 *)

A324388(n) = if(1>=omega(n),n,fordiv(n,d,if((d>1)&&(1==gcd(d,n/d)),return(n/d))));

cp's OEIS Frontend

A006022 Sprague-Grundy (or Nim) values for the game of Maundy cake on an n X 1 sheet.

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