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Definition: m < n is an extradivisor of n if for some positive k < n, m | n | k^(n+1) + m and n | (n-k)^(n+1) + m. This sequence gives the smallest number with n extradivisors.
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%I #16 Jun 19 2020 03:55:34

%S 1,2,45,105,1365,1305,4305,11445

%N Definition: m < n is an extradivisor of n if for some positive k < n, m | n | k^(n+1) + m and n | (n-k)^(n+1) + m. This sequence gives the smallest number with n extradivisors.

%e a(0) = 1 with extradivisors {};

%e a(1) = 2 with extradivisor {1};

%e a(2) = 45 with extradivisors {5, 9};

%e a(3) = 105 with extradivisors {5, 21, 35};

%e a(4) = 1365 with extradivisors {35, 105, 195, 455};

%e a(5) = 1305 with extradivisors {5, 9, 29, 45, 261}.

%t First /@ Values@ KeySort@ PositionIndex@ Table[Count[DeleteCases[Most@ Divisors@ n, d_ /; EvenQ@ d], m_ /; Total@ Boole@ Map[Function[k, And[Mod[PowerMod[k, (n + 1), n] + m, n] == 0, Mod[PowerMod[(n - k), (n + 1), n] + m, n] == 0]], Range[n - 1]] > 0], {n, 1500}] (* _Michael De Vlieger_, Dec 17 2016, Version 10 *)

%Y Cf. A272538, A279024.

%K nonn,more

%O 0,2

%A _Juri-Stepan Gerasimov_, Dec 16 2016

%E a(3)-a(7) from _Michael De Vlieger_, Dec 07 2016

%E Definition edited by _N. J. A. Sloane_, Jun 19 2020