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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: A029367 A192541 A281545 * 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 November 22 00:32 EST 2019. Contains 329383 sequences. (Running on oeis4.)