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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: A244145 A086435 A266226 * 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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