A252041 Numbers m such that m - 3 divides m^m + 3.

1, 2, 4, 5, 6, 9, 10, 85, 105, 136, 186, 262, 820, 1161, 2626, 2926, 4924, 10396, 11656, 19689, 27637, 33736, 36046, 42886, 42901, 53866, 55189, 82741, 95266, 103762, 106822, 127401, 135460, 251506, 366796, 375220, 413326, 466966, 531445, 553456, 568876

Offset: 1

Author

Juri-Stepan Gerasimov, Dec 12 2014

Keywords

• nonn

Comments

Numbers m such that (m^m + 3)/(m - 3) is an integer.

Most but not all terms are congruent to 4 modulo 6. - Robert G. Wilson v, Dec 19 2014

Note that m^m == 3^m (mod m-3). - Robert Israel, Dec 19 2014

Examples

			2 is in this sequence because (2^2 + 3)/(2 - 3) = -7 is an integer.
4 is in this sequence because (4^4 + 3)/(4 - 3) = 259 is an integer.
7 is not in the sequence because (7^7 + 3)/4 = 411773/2, which is not an integer.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..331

Crossrefs

Cf. ...............Numbers n such that x divides y, where:

...x......y....k = 0.....k = 1.....k = 2......k = 3.......

..n-k..n^n-k..A000027...A087156...A242787....A242788......

..n-k..n^n+k..A000027..see below..A249751..this sequence..

..n+k..n^n-k..A000027...A004275...A251603....A251862......

..n+k..n^n+k..A000027...A004273...A213382....A242800......

(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Dec 28 2014)

Programs

Magma

[n: n in [4..50000] | Denominator((n^n+3)/(n-3)) eq 1];

Maple

select(t -> 3 &^t + 3 mod (t-3) = 0, [1,2,$4..10^6]); # Robert Israel, Dec 19 2014

Mathematica

fQ[n_] := Mod[PowerMod[n, n, n - 3] + 3, n - 3] == 0; Select[Range@ 1000000, fQ] (* Michael De Vlieger, Dec 13 2014; modified by Robert G. Wilson v, Dec 19 2014 *)

PARI

isok(n) = (n != 3) && (Mod(n, n-3)^n  == -3); \\ Michel Marcus, Dec 13 2014

Extensions

More terms from Michel Marcus, Dec 13 2014