A357841 Smith numbers (A006753) for which the arithmetic derivative (A003415) is also a Smith number.

4, 27, 85, 121, 166, 265, 517, 526, 634, 706, 778, 913, 985, 1633, 1822, 1966, 2173, 2218, 2326, 2434, 2605, 2785, 3505, 3802, 3865, 3973, 4306, 4369, 4765, 4918, 5248, 5674, 5818, 5926, 6178, 6385, 7186, 7726, 8185, 8257, 8653, 9193, 9301, 10201, 10489, 10606

Offset: 1

Author

Marius A. Burtea, Oct 20 2022

Keywords

• nonn
• base

Examples

			4 = A006753(1) and 4' = 4, so 4 is a term.
27 = A006753(3) and 27' = 27, so 27 is a term.
85  = A006753(5) and 85' = 22 = A006753(2), so 85 is a term.
		

Crossrefs

Cf. A003415, A006753.

Programs

Magma

sm:=func; f:=func; [n:n in [2..10700]|sm(n) and sm(Floor(f(n)))];

Mathematica

digsum[n_] := Total@IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last[#]*digsum[First@#] & /@ FactorInteger[n]) == digsum[n]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[10^4], smithQ[#] && smithQ[d[#]] &] (* Amiram Eldar, Oct 21 2022 *)