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

Author

Marius A. Burtea, Nov 20 2021

Keywords

• nonn
• base

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.

Examples

			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.
		

Crossrefs

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

Programs

Magma

f:=func; a:=[]; niven:=func; [n:n in [2..520]|niven(n) and niven(Floor(f(n)))];

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 *)