login
A099303
Greatest integer x such that x' = n, or 0 if there is no such x, where x' is the arithmetic derivative of x.
10
0, 0, 4, 6, 9, 10, 15, 14, 25, 0, 35, 22, 49, 26, 55, 0, 77, 34, 91, 38, 121, 0, 143, 46, 169, 27, 187, 0, 221, 58, 247, 62, 289, 0, 323, 0, 361, 74, 391, 42, 437, 82, 403, 86, 529, 0, 551, 94, 589, 63, 667, 0, 713, 106, 703, 0, 841, 70, 899, 118, 961, 122, 943, 0, 1073, 0
OFFSET
2,3
COMMENTS
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.
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
REFERENCES
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[x=Max[Flatten[Position[d1, n]]]; If[x>-Infinity, x, 0], {n, 2, 400}]
PROG
(Python)
from sympy import factorint
def A099303(n):
for m in range(n**2>>2, 0, -1):
if sum((m*e//p for p, e in factorint(m).items())) == n:
return m
return 0 # Chai Wah Wu, Sep 12 2022
CROSSREFS
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).
Sequence in context: A200677 A189553 A189482 * A243485 A310665 A005659
KEYWORD
nonn
AUTHOR
T. D. Noe, Oct 12 2004
STATUS
approved