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A377383 Numbers k in A020487 with arithmetic derivative k' (A003415) in A020487.

%I A377383 #6 Dec 21 2024 01:03:38
%S A377383 4,256,500,625,2500,4225,11664,12800,14580,81920,250000,262144,364500,
%T A377383 531441,800000,2125764,4734976,11943936,27541504,64000000,84050000,
%U A377383 107868672,156250000,162542848,195312500,253472000,512635136,544195584,607642880,701146368,770786560
%N A377383 Numbers k in A020487 with arithmetic derivative k' (A003415) in A020487.
%C A377383 Numbers of the form m = 2^(2^(2*k - 1)) are terms. Indeed, m is a square, so it is a term in A020487, and m' = 2^(2*k - 1)*2^(2^(2*k - 1) - 1) = 2^(2^( 2*k - 1) +2*k- 2) is also a square, so it is in A020487.
%e A377383 4' = 4 = A020487(2), so 4 is a term.
%e A377383 256 = A020487(22), 256' = 1024 = A020487(48), so 256 is a term.
%t A377383 ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ahQ[n_] := Divisible[DivisorSigma[2, n], DivisorSigma[1, n]]; Select[Range[2, 10^6], ahQ[#] && ahQ[ad[#]] &] (* _Amiram Eldar_, Dec 11 2024 *)
%o A377383 (Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; ant:=func<n|IsZero(DivisorSigma(2, n) mod DivisorSigma(1, n))>; [n:n in [2..100000]|ant(n) and ant(Floor(f(n)))];
%Y A377383 Cf. A000203, A001157, A020487, A003415.
%K A377383 nonn
%O A377383 1,1
%A A377383 _Marius A. Burtea_, Dec 05 2024