A359541 -id:A359541

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A328308

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

+10
16

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 12 2019

STATUS

approved

A359542

Inverse Möbius transform of A341999, which is the characteristic function of numbers that will never reach zero when iterated with the arithmetic derivative.

+10
6

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 0, 0, 0, 4, 0, 1, 1, 2, 0, 1, 0, 4, 0, 0, 1, 3, 0, 0, 1, 4, 0, 0, 0, 2, 2, 0, 0, 6, 0, 1, 1, 3, 0, 2, 1, 4, 0, 0, 0, 5, 0, 0, 1, 5, 0, 0, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 0, 2, 0, 6, 2, 0, 0, 4, 0, 1, 1, 4, 0, 3, 1, 2, 0, 0, 1, 8, 0, 0, 1, 4, 0, 2, 0, 5, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

Question: Why there seems to be frequency bands (horizontal stripes) in the scatter plot?

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(n) = Sum_{d|n} A341999(d).

a(n) = A000005(n) - A359541(n).

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));

A341999(n) = if(!n, n, while(n>1, n = A003415checked(n)); (!n));

A359542(n) = sumdiv(n, d, A341999(d));

CROSSREFS

Cf. A000005, A328308, A341999, A359541, A359543, A359544 (positions of 0's).

KEYWORD

nonn,look

AUTHOR

Antti Karttunen, Jan 05 2023

STATUS

approved

A359543

a(n) = 1 if A359542(n) = 0, otherwise a(n) = 0.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Question: What can be said about the distribution of 0's and 1's in this sequence? Compare also to A328308 and A359546.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

a(n) = [A359542(n) == 0], where [ ] is the Iverson bracket.

a(n) = [A359541(n) == A000005(n)].

a(n) <= A328308(n).

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));

A341999(n) = if(!n, n, while(n>1, n = A003415checked(n)); (!n));

A359542(n) = sumdiv(n, d, A341999(d));

A359543(n) = (0==A359542(n));

CROSSREFS

Characteristic function of A359544.

Cf. A000005, A003415, A328308, A341999, A359541, A359542, A359546.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 05 2023

STATUS

approved

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