A098700 Numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative of x.

2, 3, 11, 17, 23, 29, 35, 37, 47, 53, 57, 65, 67, 79, 83, 89, 93, 97, 107, 117, 125, 127, 137, 145, 149, 157, 163, 173, 177, 179, 189, 197, 205, 207, 209, 217, 219, 223, 233, 237, 245, 257, 261, 277, 289, 303, 305, 307, 317, 323, 325, 337, 345, 353, 367, 373

Offset: 1

Author

Robert G. Wilson v, Sep 21 2004

Keywords

• nonn

Comments

If x' = n has solutions, they occur for x <= (n/2)^2. - T. D. Noe, Oct 12 2004

The prime and composite terms are in A189483 and A189554, respectively.

A099302(a(n)) = 0. - Reinhard Zumkeller, Mar 18 2014

Links

• T. D. Noe, Table of n, a(n) for n = 1..5000
• Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003. (See p. 7.)

Crossrefs

Cf. A003415 (arithmetic derivative of n), A099302 (number of solutions to x' = n), A099303 (greatest x such that x' = n), A098699 (least x such that x' = n).

Cf. A239433 (complement), A002620.

Subsequence of A369464.

Programs

Haskell

a098700 n = a098700_list !! (n-1)
a098700_list = filter
   (\z -> all (/= z) $ map a003415 [1 .. a002620 z]) [2..]
-- Reinhard Zumkeller, Mar 18 2014

Mathematica

a[1] = 0; a[n_] := Block[{f = Transpose[ FactorInteger[ n]]}, If[ PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; b = Table[ -1, {500}]; b[[1]] = 1; Do[c = a[n]; If[c < 500 && b[[c + 1]] == 0, b[[c + 1]] = n], {n, 10^6}]; Select[ Range[500], b[[ # ]] == 0 &]
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}]; Select[Range[400], 0==Count[d1, # ]&]

PARI

list(lim)=my(v=List()); lim\=1; forfactored(n=1, lim^2, my(f=n[2],t); listput(v, n[1]*sum(i=1, #f~, f[i, 2]/f[i, 1]))); setminus([1..lim], Set(v)); \\ Charles R Greathouse IV, Oct 21 2021

Python

from itertools import count, islice
from sympy import factorint
def A098700_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:sum((m*e//p for p,e in factorint(m).items())) != n,range(1,(n**2>>1)+1))),count(max(startvalue,2)))
A098700_list = list(islice(A098700_gen(),30)) # Chai Wah Wu, Sep 12 2022

Extensions

Corrected and extended by T. D. Noe, Oct 12 2004