%I #26 Oct 22 2017 08:15:34

%S 1,2,6,42,1806,47058,2214502422,8490421583559688410706771261086

%N Numbers n such that 1^(k*n) + 2^(k*n) + ... + (k*n)^(k*n) == k (mod k*n) for some k; that is, numbers n such that A031971(k*n) == k (mod k*n) for some k.

%C Least such k is A231409. No other terms for n < 10^110 (see Grau, Oller-Marcen, Sondow (2015) p. 428). - _Jonathan Sondow_, Nov 30 2013

%C Same as quotients Q = m/n of solutions to the congruence 1^m + 2^m + . . . + m^m == n (mod m) with n|m. For Q > 1, a necessary condition is that Q be a primary pseudoperfect number A054377. The condition is not sufficient since the primary pseudoperfect number 52495396602 is not a member. - _Jonathan Sondow_, Jul 13 2014

%H Jose María Grau, A. M. Oller-Marcen, and J. Sondow, <a href="http://arxiv.org/abs/1309.7941">On the congruence 1^m + 2^m + . . . + m^m == n (mod m) with n|m</a>, Monatshefte für Mathematik 177 (2015) 421-436, DOI 10.1007/s00605-014-0660-0

%F a(n) = A054377(n-1) for n = 2, 3, 4, 5, 6, 7, but a(8) = A054377(8). - _Jonathan Sondow_, Jul 13 2014

%Y Cf. A031971, A231409.

%Y Cf. A231562 (numbers n such that A031971(8490421583559688410706771261086*n) == n (mod 8490421583559688410706771261086*n)).

%Y Cf. A229312 (numbers n such that A031971(47058*n) == n (mod 47058*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. A229313 (numbers n such that A031971(47058*n) <> n (mod 47058*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).

%Y Cf. A054377 (primary pseudoperfect numbers).

%K nonn,more,hard

%O 1,2

%A _José María Grau Ribas_, Nov 06 2013

%E Definition corrected by _Jonathan Sondow_, Nov 30 2013