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

%I A252041 #47 Nov 12 2024 20:38:05
%S A252041 1,2,4,5,6,9,10,85,105,136,186,262,820,1161,2626,2926,4924,10396,
%T A252041 11656,19689,27637,33736,36046,42886,42901,53866,55189,82741,95266,
%U A252041 103762,106822,127401,135460,251506,366796,375220,413326,466966,531445,553456,568876
%N A252041 Numbers m such that m - 3 divides m^m + 3.
%C A252041 Numbers m such that (m^m + 3)/(m - 3) is an integer.
%C A252041 Most but not all terms are congruent to 4 modulo 6. - _Robert G. Wilson v_, Dec 19 2014
%C A252041 Note that m^m == 3^m (mod m-3). - _Robert Israel_, Dec 19 2014
%H A252041 Robert G. Wilson v, <a href="/A252041/b252041.txt">Table of n, a(n) for n = 1..331</a>
%e A252041 2 is in this sequence because (2^2 + 3)/(2 - 3) = -7 is an integer.
%e A252041 4 is in this sequence because (4^4 + 3)/(4 - 3) = 259 is an integer.
%e A252041 7 is not in the sequence because (7^7 + 3)/4 = 411773/2, which is not an integer.
%p A252041 select(t -> 3 &^t + 3 mod (t-3) = 0, [1,2,$4..10^6]); # _Robert Israel_, Dec 19 2014
%t A252041 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 *)
%o A252041 (Magma) [n: n in [4..50000] | Denominator((n^n+3)/(n-3)) eq 1];
%o A252041 (PARI) isok(n) = (n != 3) && (Mod(n, n-3)^n  == -3); \\ _Michel Marcus_, Dec 13 2014
%Y A252041 Cf. ...............Numbers n such that x divides y, where:
%Y A252041 ...x......y....k = 0.....k = 1.....k = 2......k = 3.......
%Y A252041 ..n-k..n^n-k..A000027...A087156...A242787....A242788......
%Y A252041 ..n-k..n^n+k..A000027..see below..A249751..this sequence..
%Y A252041 ..n+k..n^n-k..A000027...A004275...A251603....A251862......
%Y A252041 ..n+k..n^n+k..A000027...A004273...A213382....A242800......
%Y A252041 (For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - _David L. Harden_, Dec 28 2014)
%K A252041 nonn
%O A252041 1,2
%A A252041 _Juri-Stepan Gerasimov_, Dec 12 2014
%E A252041 More terms from _Michel Marcus_, Dec 13 2014