A099304 Least k > 0 such that (n+k)' = n' + k', where n' denotes the arithmetic derivative of n.

2, 1, 6, 2, 10, 3, 14, 4, 18, 5, 14, 6, 26, 7, 30, 8, 34, 9, 38, 10, 42, 11, 46, 12, 50, 13, 54, 14, 26, 15, 62, 16, 42, 17, 4, 18, 74, 19, 78, 20, 82, 21, 86, 22, 90, 23, 38, 24, 98, 25, 102, 26, 106, 27, 27, 28, 114, 29, 118, 30, 122, 31, 126, 32, 130, 33, 18, 34, 138, 8, 142

Offset: 1

Author

T. D. Noe, Oct 12 2004

Keywords

• nonn

Comments

The arithmetic derivative does not, in general, have the linearity property. In most cases, a(n) = n/2 for even n and a(n) = 2n for odd n.

References

• See A003415

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003415 (arithmetic derivative of n), A099305 (number of solutions to (n+k)' = n' + k').

Programs

Haskell

import Data.List (find)
import Data.Maybe (fromJust)
a099304 n = succ $ fromJust $ elemIndex 0 $
   zipWith (-) (drop (fromInteger n + 1) a003415_list)
               (map (+ n') $ tail a003415_list)
   where n' = a003415 n
-- Reinhard Zumkeller, May 09 2011

Mathematica

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[k=1; While[dn[n]+dn[k] != dn[n+k], k++ ]; k, {n, 100}]