A349484 Niven numbers whose arithmetic derivative is also a Niven number (A005349).

2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 20, 21, 27, 36, 48, 50, 54, 72, 81, 100, 108, 111, 112, 135, 153, 156, 180, 192, 201, 209, 210, 216, 224, 225, 230, 243, 280, 288, 306, 324, 336, 351, 364, 378, 392, 400, 405, 407, 420, 432, 441, 480, 481, 486, 500, 504, 511, 512

Offset

1,1

Comments

The sequence is infinite because the numbers of the form m = 2*10^(10^k), k >= 1, are terms. Indeed, m is a Niven number, m' = 10^(10^k) + 2*10^k*10^(10^k - 1)*7 = 10^(10^k - 1)*(10 + 140*10^k) = 10^(10^k)*(1 + 14*10^k), digsum(m') = 6 and m' is divisible by 6, so it is a Niven number.

Links

Table of n, a(n) for n=1..58.

Example

2 = A005349(2) and 2' = 1 = A005349(1), so 2 is a term.
18 = A005349(12) and 18' = 21 = A005349(14), so 18 is a term.

Mathematica

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[2, 512], And @@ nivenQ /@ {#, d[#]} &] (* Amiram Eldar, Nov 20 2021 *)

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)]])>; a:=[]; niven:=func<n|n mod &+Intseq(n) eq 0>; [n:n in [2..520]|niven(n) and niven(Floor(f(n)))];

Crossrefs

Cf. A002808, A005349 (Niven numbers), A003415 (arithmetic derivative).

Sequence in context: A032345 A023765 A032906 * A075905 A321767 A377477

Adjacent sequences: A349481 A349482 A349483 * A349485 A349486 A349487

Keyword

nonn,base

Author

Marius A. Burtea, Nov 20 2021

Status

approved