A279680 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.
1, 2, 45, 105, 1365, 1305, 4305, 11445
Offset: 0
Examples
a(0) = 1 with extradivisors {};
a(1) = 2 with extradivisor {1};
a(2) = 45 with extradivisors {5, 9};
a(3) = 105 with extradivisors {5, 21, 35};
a(4) = 1365 with extradivisors {35, 105, 195, 455};
a(5) = 1305 with extradivisors {5, 9, 29, 45, 261}.
Programs
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Mathematica
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 *)
Extensions
a(3)-a(7) from Michael De Vlieger, Dec 07 2016
Definition edited by N. J. A. Sloane, Jun 19 2020