A252594 - OEIS

%I #16 Dec 24 2014 23:00:38

%S 1,2,8,16,27,32,54,64,100,128,200,243,256,400,486,500,512,800,972,

%T 1000,1024,1600,1944,2000,2048,3200,3888,4000,4096,4624,6400,7776,

%U 8000,8192,9248,12100,12500,12800,15552,16000,16384,18496,24200,25000,25600,31104,32000,32768,36992,48400,50000

%N Records in A072994.

%C Consider the function f(n) = the number of positive integers k < n such that k^n (mod n) == 1. This sequence lists the values of n at which f(n) reaches a new maximum.

%C All powers of two are present except its square. f(2^n) (with exception noted) = 2^(n-1) = 2^n/2.

%C All powers of two multiplied by 100, 1000 and 100000, but not 10000, are also present.

%C Terms other than the above are 27, 54, 243, 486, 500, 972, 1944, 3888, 4624, 7776, 9248, 12100, 12500, 15552, 18496, 24200, 25000, 31104, 36992, 48400, 50000, ..., .

%C Conjecture: f(x)/x -> 5/12.

%H Robert G. Wilson v, <a href="/A252594/b252594.txt">Table of n, a(n) for n = 1..94</a>

%t f[n_] := (d = If[ OddQ@ n, 1, 2];   d*Length@ Select[ Range[ n/d], PowerMod[#, n, n] == 1 &]); f[1] = 1; k = 1; mx = 0; lst = {}; While[k < 10000001, a = f@ k; If[a > mx, mx = a; AppendTo[lst, k]; Print[{a, k}]]; k++]; lst

%Y Cf. A072994.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Dec 18 2014