A328308 a(n) = 1 if k-th arithmetic derivative of n is zero for some k, otherwise 0.

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1

Offset

0

Comments

Question: What is the asymptotic mean of this and related sequences like A368915? Compare also to A341996, A359543 and A359546.

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Michael De Vlieger, Bitmap of a(n), n = 0..2^24, 2048 X 2048 pixels, with 0 in white and 1 in black. Furnishes 4260302 terms of A099308.

Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seq., Vol. 6 (2003), Article 03.3.4.

Index entries for characteristic functions

Formula

For prime p, a(p) = 1, a(p^p * m) = 0, for all m >= 1. a(4m) = 0 for m > 0. - Michael De Vlieger, Jan 04 2023

From Antti Karttunen, Jan 06 2023: (Start)

a(0) = 1; and for n > 0, a(n) = A359550(n) * a(A003415(n)). [Provided that Conjecture 3 given on page 5 of Ufnarovski and Åhlander 2003 paper holds, i.e., that map x -> x' never forms nontrivial loops]

a(n) = 1 - A341999(n).

a(n) >= A359543(n).

(End)

For all n > 1, a(n) <= A368915(n) <= A359550(n). - Antti Karttunen, Jan 10 2024

Mathematica

w = {}; nn = 2^10; k = 1; While[Set[m, #^#] <= nn &[Prime[k]], AppendTo[w, m]; k++]; a3415[n_] := a3415[n] = Which[Abs@ n < 2, 0, PrimeQ[n], 1, True, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]{1, 1}~Join~Reap[Do[Which[PrimeQ[n], Sow[1], MemberQ[w, n], Sow[0], True, If[NestWhileList[a3415, n, And[! Divisible[#, 4], FreeQ[w, #]] &, 1][[-1]] == 0, Sow[1], Sow[0]]], {n, 2, nn}]][[-1, -1]] (* Michael De Vlieger, Jan 04 2023 *)
(* 2nd program: generate m <= 2^24 terms of the sequence from the bitmap above: *)
m = 10^3; Flatten[ImageData[Import["https://oeis.org/A328308/a328308.png"], "Bit"]][[1 ;; m]] /. {0 -> 1, 1 -> 0} (* Michael De Vlieger, Jan 04 2023 *)

Prog

(PARI)
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));
A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));

Crossrefs

Characteristic function of A099308.

Cf. A003415, A099309 (positions of zeros), A256750, A328306 [= a(A276086(n))], A328309 (partial sums), A341996, A341999 (one's complement), A342023, A351071, A359541 (inverse Möbius transform), A359543, A359546, A359550, A368915.

Sequence in context: A135839 A071022 A155076 * A257196 A176137 A290808

Adjacent sequences: A328305 A328306 A328307 * A328309 A328310 A328311

Keyword

nonn

Author

Antti Karttunen, Oct 12 2019

Status

approved