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A380393 a(n) is the least k that has exactly n proper divisors d such that (-d)^k == -d (mod k).

%I A380393 #18 Mar 31 2025 23:09:51
%S A380393 1,2,6,42,66,105,2805,561,1365,5005,5565,11305,36465,140505,239785,
%T A380393 41041,682465,873145,185185,418285,1683969,2113665,5503785,1242241,
%U A380393 6697405,8549905,31932901,11996985,31260405,30534805,47031061,825265,27265161,32306365,55336645,21662641,9276085,8964865
%N A380393 a(n) is the least k that has exactly n proper divisors d such that (-d)^k == -d (mod k).
%C A380393 a(n) is the least k such that A378387(k) = n.
%C A380393 It appears that a(n) = A371513(n) except for n = 4 and n = 6. They are certainly equal when a(n) and A371513(n) are odd, since if k is odd, (-d)^k = -d^k == -d (mod k) if and only if d^k == d (mod k).
%F A380393 A378387(a(n)) = n.
%e A380393 a(4) = 42 because 42 has 3 such divisors: (-6)^42 == 36 == -6 (mod 42), (-14)^42 == 28 == -14 (mod 42), (-21)^42 == 21 == -21 (mod 42), and no smaller number works.
%p A380393 f:= proc(n) nops(select((t -> (-t)&^n + t mod n = 0), numtheory:-divisors(n) minus {n})) end proc:
%p A380393 N:= 30: # for a(0) .. a(N)
%p A380393 V:= Array(0..N): count:= 0:
%p A380393 for i from 1 while count < N+1 do
%p A380393   v:= f(i);
%p A380393   if v <= N and V[v] = 0 then V[v]:= i; count:= count+1 fi;
%p A380393 od:
%p A380393 convert(V,list);
%o A380393 (PARI) a(n) = my(k=1); while (sumdiv(k, d, if (d<k, my(x=Mod(-d, k)); x^k == x)) != n, k++); k; \\ _Michel Marcus_, Jan 24 2025
%Y A380393 Cf. A371513, A378387.
%K A380393 nonn
%O A380393 0,2
%A A380393 _Robert Israel_, Jan 23 2025