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

1,1

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A076891 A071883 A368241 * A064680 A352544 A363596

Adjacent sequences: A099301 A099302 A099303 * A099305 A099306 A099307

Keyword

nonn

Author

T. D. Noe, Oct 12 2004

Status

approved