

A099308


Numbers m whose kth arithmetic derivative is zero for some k. Complement of A099309.


24



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, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 105, 107, 109, 113, 114, 118, 121, 126, 127, 129, 130
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OFFSET

1,3


COMMENTS

The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.
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


REFERENCES

See A003415


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
Victor Ufnarovski and Bo Ã…hlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.


FORMULA

For all n >= 0, A328309(a(n)) = n.  Antti Karttunen, Feb 14 2022


EXAMPLE

18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 0.


MATHEMATICA

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]


PROG

(PARI)
\\ The following program would get stuck in nontrivial loops. However, we assume that the conjecture 3 in Ufnarovski & Ã…hlander paper holds ("The differential equation n (k) = n has only trivial solutions p p for primes p").
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));
isA099308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n)); \\ Antti Karttunen, Feb 14 2022


CROSSREFS

Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the kth arithmetic derivative of n is zero), A099309 (numbers whose kth 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).
Cf. A328308, A328309 (characteristic function and their partial sums), A341999 (1  charfun).
Cf. A276086, A328116, A351255, A351257, A351259, A351261, A351072 (number of prime(k)smooth terms > 1).
Cf. also A256750, A327969, A351088.
Sequence in context: A305847 A248565 A065896 * A074235 A325366 A192189
Adjacent sequences: A099305 A099306 A099307 * A099309 A099310 A099311


KEYWORD

nonn


AUTHOR

T. D. Noe, Oct 12 2004


STATUS

approved



