Owen C. Keith - cp's OEIS frontend

User: Owen C. Keith

Owen C. Keith has authored 2 sequences.

A342144 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with x > 0.

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 155, 157, 159, 161, 163, 167

Offset: 1

Owen C. Keith, Mar 01 2021

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.
First differs from A334420 at a(70) which is 167 for this sequence and 165 for A334420.
First differs from A056911 at a(21) which is 49 for this sequence and 51 for A056911.

			3 is a term since 1^1 == 2^2 (mod 3).
5 is a term since 11^11 == 12^12 (mod 5).

Cf. A056911, A084820, A174824, A334420, A338445.

seqQ[n_] := AnyTrue[Range[LCM[n, CarmichaelLambda[n]]+1], PowerMod[#, #, n] == PowerMod[# + 1, # + 1, n] &]; Select[Range[145], seqQ]

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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User: Owen C. Keith

A342144 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with x > 0.

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