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| A230311
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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.
(history;
published version)
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#26 by Alois P. Heinz at Sun Oct 22 08:15:34 EDT 2017
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#25 by Jonathan Sondow at Sun Oct 22 08:12:24 EDT 2017
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#24 by Jonathan Sondow at Sun Oct 22 08:11:57 EDT 2017
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| COMMENTS
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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
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| LINKS
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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
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| STATUS
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approved
editing
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#23 by Peter Luschny at Sun Oct 22 07:49:28 EDT 2017
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#22 by Jonathan Sondow at Sun Oct 22 07:44:05 EDT 2017
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#21 by Jonathan Sondow at Sun Oct 22 07:43:32 EDT 2017
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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
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| LINKS
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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
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approved
editing
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#20 by Michel Marcus at Sun Jul 13 12:48:05 EDT 2014
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#19 by Joerg Arndt at Sun Jul 13 12:39:08 EDT 2014
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#18 by Jonathan Sondow at Sun Jul 13 12:12:55 EDT 2014
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Discussion
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| Sun Jul 13
| 12:39
| Joerg Arndt: Thanks!
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#17 by Jonathan Sondow at Sun Jul 13 12:12:26 EDT 2014
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| LINKS
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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, DOI 10.1007/s00605-014-0660-0
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Discussion
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| Sun Jul 13
| 12:12
| Jonathan Sondow: I updated the title of the link.
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