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A099308 Numbers m whose k-th 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 (list; graph; refs; listen; history; text; internal format)
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 k-th arithmetic derivative of n is zero), A099309 (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).

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

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Last modified September 27 09:05 EDT 2022. Contains 357054 sequences. (Running on oeis4.)