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 A099302 Number of integer solutions to x' = n, where x' is the arithmetic derivative of x. 18
 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 3, 0, 2, 1, 2, 2, 3, 0, 4, 1, 3, 1, 2, 0, 3, 2, 4, 1, 4, 0, 4, 0, 2, 2, 3, 1, 4, 1, 4, 2, 4, 0, 6, 1, 4, 1, 3, 0, 5, 2, 4, 0, 4, 1, 7, 2, 3, 1, 5, 0, 6, 0, 3, 1, 5, 2, 7, 1, 5, 3, 5, 1, 7, 0, 6, 2, 5, 0, 8, 1, 5, 2, 4, 0, 9, 3, 6, 0, 5, 1, 8, 0, 3, 1, 6, 2, 8, 2, 5, 1, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,9 COMMENTS This is the i(n) function in the paper by Ufnarovski and Ahlander. Note that a(1) is infinite because all primes satisfy x' = 1. The plot shows the great difference in the number of solutions for even and odd n. Also compare sequence A189558, which gives the least number have n solutions, and A189560, which gives the least such odd number. It appears that there are a finite number of even numbers having a given number of solutions. This conjecture is explored in A189561 and A189562. a(A098700(n)) = 0; a(A239433(n)) > 0. - Reinhard Zumkeller, Mar 18 2014 REFERENCES See A003415 LINKS T. D. Noe, Table of n, a(n) for n = 2..40000 Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003. MATHEMATICA dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; Table[Count[d1, n], {n, 2, 400}] PROG (Haskell) a099302 n = length \$ filter (== n) \$ map a003415 [1 .. a002620 n] -- Reinhard Zumkeller, Mar 18 2014 CROSSREFS Cf. A003415 (arithmetic derivative of n), A099303 (greatest x such that x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution). Cf. A002620. Sequence in context: A281545 A332033 A331981 * A219198 A025814 A029354 Adjacent sequences:  A099299 A099300 A099301 * A099303 A099304 A099305 KEYWORD nonn,look AUTHOR T. D. Noe, Oct 12 2004, Apr 24 2011 STATUS approved

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Last modified July 30 23:12 EDT 2021. Contains 346365 sequences. (Running on oeis4.)