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

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

See A003415

Links

T. D. Noe, Table of n, a(n) for n=2..1000

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

Adjacent sequences: A099300 A099301 A099302 * A099304 A099305 A099306

Keyword

nonn

Author

T. D. Noe, Oct 12 2004

Status

approved