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A384684 Nonprimes k such that sopf(k)^k == sopf(k) (mod k) where sopf = A008472.

%I A384684 #24 Jul 07 2025 14:26:06
%S A384684 1,28,30,45,65,66,90,105,133,190,231,286,301,325,369,385,426,496,532,
%T A384684 561,645,793,946,1016,1105,1288,1353,1729,1905,2041,2107,2121,2275,
%U A384684 2278,2413,2465,2501,2701,2737,2821,3577,3781,3861,4015,4123,4161,4699
%N A384684 Nonprimes k such that sopf(k)^k == sopf(k) (mod k) where sopf = A008472.
%C A384684 Superset of even perfect numbers > 6.
%H A384684 Robert Israel, <a href="/A384684/b384684.txt">Table of n, a(n) for n = 1..10000</a>
%p A384684 filter:= proc(k) local s;
%p A384684   if isprime(k) then return false fi;
%p A384684   s:= convert(numtheory:-factorset(k),`+`);
%p A384684   (s &^ k - s) mod k = 0;
%p A384684 end proc:
%p A384684 select(filter, [$1..10000]); # _Robert Israel_, Jun 12 2025
%t A384684 A008472[n_]:=Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Select[Range[5000],PowerMod[A008472[#],#,#]==A008472[#] &] (* _Stefano Spezia_, Jun 07 2025 *)
%o A384684 (Magma) [k: k in [1..5000] | not IsPrime(k) and Modexp(k eq 1 select 0 else &+[p[1]: p in Factorization(k)], k, k) eq (k eq 1 select 0 else &+[p[1]: p in Factorization(k)])];
%o A384684 (PARI) isok(k) = if (!isprime(k), my(s=vecsum(factor(k)[, 1])); Mod(s, k)^k == s); \\ _Michel Marcus_, Jun 12 2025
%Y A384684 Cf. A000396, A005835, A008472, A107290, A239546, A302333.
%K A384684 nonn
%O A384684 1,2
%A A384684 _Juri-Stepan Gerasimov_, Jun 06 2025