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Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.
10

%I #17 Mar 12 2023 14:03:32

%S 0,0,4,6,9,10,15,14,25,0,35,22,49,26,55,0,77,34,91,38,121,0,143,46,

%T 169,27,187,0,221,58,247,62,289,0,323,0,361,74,391,42,437,82,403,86,

%U 529,0,551,94,589,63,667,0,713,106,703,0,841,70,899,118,961,122,943,0,1073,0

%N Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

%C This is the largest member of the set I(n) in the paper by Ufnarovski and Ahlander. They show that a(n) <= (n/2)^2.

%C Because this sequence is quite different for even and odd n, it is bisected into A102084 and A189762. The upper bound for odd n appears to be (n/3)^(3/2), which is attained when n = 3p^2 for primes p>5. - _T. D. Noe_, Apr 27 2011

%D See A003415

%H T. D. Noe, <a href="/A099303/b099303.txt">Table of n, a(n) for n=2..1000</a>

%t 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[x=Max[Flatten[Position[d1, n]]]; If[x>-Infinity, x, 0], {n, 2, 400}]

%o (Python)

%o from sympy import factorint

%o def A099303(n):

%o for m in range(n**2>>2,0,-1):

%o if sum((m*e//p for p,e in factorint(m).items())) == n:

%o return m

%o return 0 # _Chai Wah Wu_, Sep 12 2022

%Y Cf. A003415 (arithmetic derivative of n), A099302 (number of solutions to x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution).

%K nonn

%O 2,3

%A _T. D. Noe_, Oct 12 2004