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

1,1

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.

Links

Table of n, a(n) for n=1..31.

Example

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

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 *)

Prog

(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)))];

Crossrefs

Cf. A000203, A001157, A020487, A003415.

Sequence in context: A346919 A203511 A063073 * A093760 A062075 A266890

Adjacent sequences: A377380 A377381 A377382 * A377384 A377385 A377386

Keyword

nonn

Author

Marius A. Burtea, Dec 05 2024

Status

approved