A230311 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.

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

Offset

1,2

Comments

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

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

Links

Table of n, a(n) for n=1..8.

Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + . . . + m^m == n (mod m) with n|m, Monatshefte für Mathematik 177 (2015) 421-436, DOI 10.1007/s00605-014-0660-0

Formula

a(n) = A054377(n-1) for n = 2, 3, 4, 5, 6, 7, but a(8) = A054377(8). - Jonathan Sondow, Jul 13 2014

Crossrefs

Cf. A031971, A231409.

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

Cf. A229312 (numbers n such that A031971(47058*n) == n (mod 47058*n)).

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

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

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

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

Cf. A229313 (numbers n such that A031971(47058*n) <> n (mod 47058*n)).

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

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

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

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

Cf. A229308 (primitive numbers in A229304).

Cf. A229309 (primitive numbers in A229305).

Cf. A229310 (primitive numbers in A229306).

Cf. A229311 (primitive numbers in A229307).

Cf. A054377 (primary pseudoperfect numbers).

Sequence in context: A242927 A054377 A349193 * A276416 A007018 A100016

Adjacent sequences: A230308 A230309 A230310 * A230312 A230313 A230314

Keyword

nonn,more,hard

Author

José María Grau Ribas, Nov 06 2013

Extensions

Definition corrected by Jonathan Sondow, Nov 30 2013

Status

approved