A099305 Number of solutions of the equation (n+k)' = n' + k', with 1 <= k <= 2n, where n' denotes the arithmetic derivative of n.

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 1, 2, 3, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 1, 3, 3, 3, 1, 2, 2, 3, 2

Offset

1,2

Comments

Observe that when n and c*n have the same parity, a(c*n) >= a(n) for all integers c. For even n, there are always at least two solutions, k=n/2 and k=2n. For odd n, k=2n is always a solution.

a(A258138(n)) = n and a(m) != n for m < A258138(n). - Reinhard Zumkeller, May 21 2015

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[lst={}; k=0; While[k<2n, k++; While[k<=2n && dn[n]+dn[k] != dn[n+k], k++ ]; If[dn[n]+dn[k]==dn[n+k], AppendTo[lst, k]]]; Length[lst], {n, 100}]

Prog

(Haskell)
a099305 n = a099305_list !! (n-1)
a099305_list = f 1 $ h 1 empty where
   f x ad = y : f (x + 1) (h (3 * x + 1) ad)  where
            y = length [() | k <- [1 .. 2 * x],
                             let x' = ad ! x, ad ! (x + k) == x' + ad ! k]
   h z = insert z (a003415 z) .
          insert (z+1) (a003415 (z+1)) . insert (z+2) (a003415 (z+2))
-- Reinhard Zumkeller, May 21 2015

Crossrefs

Cf. A003415 (arithmetic derivative of n), A099304 (least k > 0 such that (n+k)' = n' + k').

Cf. A258138.

Sequence in context: A086435 A266226 A376885 * A334461 A338652 A033109

Adjacent sequences: A099302 A099303 A099304 * A099306 A099307 A099308

Keyword

nonn

Author

T. D. Noe, Oct 12 2004

Status

approved