cp's OEIS Frontend

A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.

%I A099303 #17 Mar 12 2023 14:03:32
%S A099303 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 A099303 169,27,187,0,221,58,247,62,289,0,323,0,361,74,391,42,437,82,403,86,
%U A099303 529,0,551,94,589,63,667,0,713,106,703,0,841,70,899,118,961,122,943,0,1073,0
%N A099303 Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.
%C A099303 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 A099303 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 A099303 See A003415
%H A099303 T. D. Noe, <a href="/A099303/b099303.txt">Table of n, a(n) for n=2..1000</a>
%t A099303 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 A099303 (Python)
%o A099303 from sympy import factorint
%o A099303 def A099303(n):
%o A099303     for m in range(n**2>>2,0,-1):
%o A099303         if sum((m*e//p for p,e in factorint(m).items())) == n:
%o A099303             return m
%o A099303     return 0 # _Chai Wah Wu_, Sep 12 2022
%Y A099303 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 A099303 nonn
%O A099303 2,3
%A A099303 _T. D. Noe_, Oct 12 2004