A239061 -id:A239061 - cp's OEIS frontend

A239063 Numbers k for which the equation x^x == 1 (mod k) has x=1 as its only solution such that 1 <= x <= k.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 32, 34, 36, 40, 42, 46, 48, 50, 54, 60, 64, 68, 72, 78, 80, 82, 84, 90, 94, 96, 98, 100, 102, 108, 110, 114, 120, 126, 128, 136, 144, 150, 156, 160, 162, 164, 168, 170, 176, 180, 186, 188, 192, 194, 198, 200, 204

Offset: 1

José María Grau Ribas, Mar 09 2014

nonn

A number k belongs to this sequence if and only if for all primes p|k and for all primes q|p-1, q divides k. - Bernat Pagès Vives, Oct 30 2021

Giovanni Resta, Table of n, a(n) for n = 1..10000
Art of Problem Solving, OMO Fall 2013 Solutions, problem 22

Cf. A239061.

gg1[n_] := Sum[If[Mod[x^x, n] == Mod[1, n], 1, 0],{x,1,n}]; Select[Range[100], gg1[#] == 1 &]

A239062 Number of integers x, 1 <= x <= n, such that x^x == 0 (mod n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 7, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 15, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 31, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 8, 26, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 10, 1, 1, 1, 4, 1

Offset: 1

José María Grau Ribas, Mar 09 2014

nonn

			From _Michael De Vlieger_, Sep 23 2017: (Start)
Table of records a(n) and first positions n:
   i       n    a(n)
-------------------
   1       1      1
   2       4      2
   3       8      3
   4      16      7
   5      27      9
   6      32     15
   7      64     31
   8     128     62
   9     243     80
  10     256    126
  11     512    253
  12    1024    509
  13    2048   1020
  14    4096   2044
  15    6561   2185
  16    8192   4092
  17   16384   8188
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A239061, A239063, A005117 (indices of 1's).

gg0[n_] := Sum[If[Mod[x^x , n] == 0, 1, 0], {x, n}];Array[gg0,200]
(* or *)
Array[Sum[Boole[PowerMod[x, x, #] == 0], {x, #}] &, 10^4] (* or *)
Table[Count[Range@ n, k_ /; PowerMod[k, k, n] == 0], {n, 200}] (* Michael De Vlieger, Sep 23 2017 *)

A239062(n) = sum(x=1,n, if(0 == Mod(x^x, n), 1, 0)); \\ Antti Karttunen, Sep 23 2017, after the Mathematica-program.

More terms from Antti Karttunen, Sep 23 2017

A338445 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with 1<=x

Original entry on oeis.org

3, 11, 13, 19, 23, 29, 31, 43, 49, 53, 57, 59, 61, 67, 71, 73, 77, 79, 83, 85, 89, 91, 93, 97, 101, 103, 109, 113, 127, 129, 131, 133, 141, 143, 147, 149, 151, 157, 161, 163, 167, 169, 173, 177, 179, 183, 187, 197, 199, 201, 203, 205, 211, 217, 229, 235, 237, 239

Offset: 1

Owen C. Keith, Oct 28 2020

nonn

Some values of m have multiple solutions.
For example, for m = 49, 25^25 == 26^26 (mod 49) and 37^37 == 38^38 (mod 49).
All terms are odd. - Robert Israel, Nov 25 2020

			3 is a term because 1^1 == 2^2 (mod 3).
11 is a term because 8^8 == 9^9 (mod 11).
13 is a term because 8^8 == 9^9 (mod 13).

Robert Israel, Table of n, a(n) for n = 1..10000

Similar sequences: A174824, A239061, A239062, A239063.

filter:= proc(n) local x,y,z;
  y:= 1;
  for x from 2 to n-1 do
    z:= x &^ x mod n;
    if z = y then return true fi;
    y:= z
  od;
  false
end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 25 2020

seqQ[n_] := AnyTrue[Range[n - 1], PowerMod[#, #, n] == PowerMod[# + 1, # + 1, n] &]; Select[Range[240], seqQ] (* Amiram Eldar, Oct 28 2020 *)

isok(m)=sum(i=1, m-1, Mod(i,m)^i == Mod((i+1),m)^(i+1)) \\ Andrew Howroyd, Oct 28 2020

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A239063 Numbers k for which the equation x^x == 1 (mod k) has x=1 as its only solution such that 1 <= x <= k.

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A239062 Number of integers x, 1 <= x <= n, such that x^x == 0 (mod n).

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A338445 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with 1<=x

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