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 A328308 a(n) = 1 if k-th arithmetic derivative of n is zero for some k, otherwise 0. 13
 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 0 COMMENTS Question: What can be said about the distribution of 0's and 1's in this sequence? 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. 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)). a(n) = 1 - A341999(n). a(n) >= A359543(n). (End) 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. 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

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Last modified September 26 15:12 EDT 2023. Contains 365660 sequences. (Running on oeis4.)