A098700 - OEIS

%I #42 Jan 24 2024 13:54:52

%S 2,3,11,17,23,29,35,37,47,53,57,65,67,79,83,89,93,97,107,117,125,127,

%T 137,145,149,157,163,173,177,179,189,197,205,207,209,217,219,223,233,

%U 237,245,257,261,277,289,303,305,307,317,323,325,337,345,353,367,373

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

%C If x' = n has solutions, they occur for x <= (n/2)^2. - _T. D. Noe_, Oct 12 2004

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

%C A099302(a(n)) = 0. - _Reinhard Zumkeller_, Mar 18 2014

%H T. D. Noe, <a href="/A098700/b098700.txt">Table of n, a(n) for n = 1..5000</a>

%H Victor Ufnarovski and Bo Ahlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003. (See p. 7.)

%t 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 &]

%t 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, # ]&]

%o (Haskell)

%o a098700 n = a098700_list !! (n-1)

%o a098700_list = filter

%o    (\z -> all (/= z) $ map a003415 [1 .. a002620 z]) [2..]

%o -- _Reinhard Zumkeller_, Mar 18 2014

%o (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

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o def A098700_gen(startvalue=2): # generator of terms >= startvalue

%o     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)))

%o A098700_list = list(islice(A098700_gen(),30)) # _Chai Wah Wu_, Sep 12 2022

%Y 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).

%Y Cf. A239433 (complement), A002620.

%Y Subsequence of A369464.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Sep 21 2004

%E Corrected and extended by _T. D. Noe_, Oct 12 2004