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

1,1

Links

Table of n, a(n) for n=1..46.

Example

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.

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 *)

Prog

(Magma) sm:=func<n| not IsPrime(n) and (&+Intseq(n) eq &+[  Factorisation(n)[i][2]* &+Intseq(Factorisation(n)[i][1]) : i in [1..#PrimeDivisors(n)]])>; f:=func<h |h le 1 select 0 else  h*(&+[Factorisation(h)[i][2] / Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n:n in [2..10700]|sm(n) and sm(Floor(f(n)))];

Crossrefs

Cf. A003415, A006753.

Sequence in context: A015238 A298987 A070600 * A100488 A220019 A225902

Adjacent sequences: A357838 A357839 A357840 * A357842 A357843 A357844

Keyword

nonn,base

Author

Marius A. Burtea, Oct 20 2022

Status

approved