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%I #31 Jan 06 2023 15:23:42
%S 0,1,2,3,5,6,7,9,10,11,13,14,17,18,19,21,22,23,25,29,30,31,33,34,37,
%T 38,41,42,43,46,47,49,53,57,58,59,61,62,65,66,67,70,71,73,77,78,79,82,
%U 83,85,89,93,94,97,98,101,103,105,107,109,113,114,118,121,126,127,129,130
%N Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.
%C The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.
%C For all n, A003415(a(n)) is also a term of the sequence. A351255 gives the nonzero terms as ordered by their position in A276086. - _Antti Karttunen_, Feb 14 2022
%D See A003415.
%H T. D. Noe, <a href="/A099308/b099308.txt">Table of n, a(n) for n = 1..10000</a>
%H Victor Ufnarovski and Bo Åhlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
%F For all n >= 0, A328309(a(n)) = n. - _Antti Karttunen_, Feb 14 2022
%e 18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 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}]; nLim=200; lst={1}; i=1; While[i<=Length[lst], currN=lst[[i]]; pre=Intersection[Flatten[Position[d1, currN]], Range[nLim]]; pre=Complement[pre, lst]; lst=Join[lst, pre]; i++ ]; Union[lst]
%o (PARI)
%o \\ The following program would get stuck in nontrivial loops. However, we assume that the conjecture 3 in Ufnarovski & Åhlander paper holds ("The differential equation n^(k) = n has only trivial solutions p^p for primes p").
%o A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
%o isA099308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n)); \\ _Antti Karttunen_, Feb 14 2022
%Y Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099309 (complement, numbers whose k-th arithmetic derivative is nonzero for all k), A351078 (first noncomposite reached when iterating the derivative from these numbers), A351079 (the largest term on such paths).
%Y Cf. A328308, A328309 (characteristic function and their partial sums), A341999 (1 - charfun).
%Y Cf. A276086, A328116, A351255 (permutation of nonzero terms), A351257, A351259, A351261, A351072 (number of prime(k)-smooth terms > 1).
%Y Cf. also A256750 (number of iterations needed to reach either 0 or a number with a factor of the form p^p), A327969, A351088.
%Y Union of A359544 and A359545.
%K nonn
%O 1,3
%A _T. D. Noe_, Oct 12 2004
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