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A099305 Number of solutions of the equation (n+k)' = n' + k', with 1 <= k <= 2n, where n' denotes the arithmetic derivative of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A214715 A244145 A086435 * A033109 A235644 A175096

Adjacent sequences:  A099302 A099303 A099304 * A099306 A099307 A099308

KEYWORD

nonn

AUTHOR

T. D. Noe, Oct 12 2004

STATUS

approved

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Last modified July 5 02:05 EDT 2015. Contains 259225 sequences.