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

1,1

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.)

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, # ]&]

Prog

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

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.

Sequence in context: A014092 A100962 A045337 * A025584 A242256 A189483

Adjacent sequences: A098697 A098698 A098699 * A098701 A098702 A098703

Keyword

nonn

Author

Robert G. Wilson v, Sep 21 2004

Extensions

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

Status

approved