A229307 - OEIS

%I #23 Sep 30 2020 03:29:46

%S 3,6,9,10,12,15,18,20,21,24,27,30,33,36,39,40,42,45,48,50,51,54,55,57,

%T 60,63,66,69,70,72,75,78,80,81,84,87,90,93,96,99,100,102,105,108,110,

%U 111,114,117,120,123,126,129,130,132,135,136,138,140,141,144

%N Numbers k such that A031971(2*k) <> k (mod 2*k).

%C Complement of A229303.

%C The asymptotic density is in [0.41545, 0.416846].

%C If n is in A then k*n is in A for all natural number k.

%C The numbers k = 1, 2, 6, 42, 1806, 47058, 2214502422, 8490421583559688410706771261086 = A230311 are the only values of k such that the set {n: A031971(k*n) == n (mod k*n)} is nonempty. Its smallest element is n = 1, 1, 1, 1, 1, 5, 5, 39607528021345872635 = A231409. [Comment corrected and expanded by _Jonathan Sondow_, Dec 10 2013]

%H Amiram Eldar, <a href="/A229307/b229307.txt">Table of n, a(n) for n = 1..10000</a>

%H José María Grau, Antonio M. Oller-Marcén and Jonathan Sondow, <a href="https://doi.org/10.1007/s00605-014-0660-0">On the congruence 1^m + 2^m + ... + m^m = n (mod m), with n|m</a>, Monatshefte für Mathematik, Vol. 177, No. 3 (2015), pp. 421-436, <a href="https://arxiv.org/abs/1309.7941">preprint</a>, arXiv:1309.7941 [math.NT], 2013-2014.

%t g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[500], !g[2*#] == # &]

%Y Cf. A031971.

%Y Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).

%Y Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).

%Y Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).

%Y Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).

%Y Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).

%Y Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).

%Y Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).

%Y Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).

%Y Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).

%Y Cf. A229308 (primitive numbers in A229304).

%Y Cf. A229309 (primitive numbers in A229305).

%Y Cf. A229310 (primitive numbers in A229306).

%Y Cf. A229311 (primitive numbers in A229307).

%K nonn

%O 1,1

%A _José María Grau Ribas_, Sep 24 2013