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

4, 256, 500, 625, 2500, 4225, 11664, 12800, 14580, 81920, 250000, 262144, 364500, 531441, 800000, 2125764, 4734976, 11943936, 27541504, 64000000, 84050000, 107868672, 156250000, 162542848, 195312500, 253472000, 512635136, 544195584, 607642880, 701146368, 770786560

Offset: 1

Author

Marius A. Burtea, Dec 05 2024

Keywords

• nonn

Comments

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.

Examples

			4' = 4 = A020487(2), so 4 is a term.
256 = A020487(22), 256' = 1024 = A020487(48), so 256 is a term.
		

Crossrefs

Cf. A000203, A001157, A020487, A003415.

Programs

Magma

f:=func; ant:=func; [n:n in [2..100000]|ant(n) and ant(Floor(f(n)))];

Mathematica

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 *)