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A098700
Numbers n such that x' = n has no integer solution, where x' is the arithmetic derivative of x.
14
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
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
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Sep 21 2004
EXTENSIONS
Corrected and extended by T. D. Noe, Oct 12 2004
STATUS
approved