This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A189760 #27 Oct 02 2019 20:06:11
%S A189760 0,1,2,6,9,14,33,62,177,414,1155,1719,2625,4018,6849,9770,17675,30206,
%T A189760 90609,260343,336006,757995,1290874,2029875,4059746,7037655,17594075,
%U A189760 50850483,68589598,186888243,373659254,1884639669
%N A189760 Least nonnegative number whose n-th arithmetic derivative (A003415) is zero and lower derivatives are nonzero.
%C A189760 a(32) <= 9519378185. - _Donovan Johnson_, Apr 30 2011
%C A189760 From _Antti Karttunen_, Oct 02 2019: (Start)
%C A189760 For at least n = 1, 3, 4, 5, 6, 7, 10, 14, 15, 17, 21, 23, 24, 25, 26, 27, 28, 29, we have = a(n) = A003415(a(1+n)), thus we have subsequences like 6, 9, 14, 33, 62, 177 that are obtained by iterating A098699 starting from 6, but as A098699(177) = 0, that run ends there. From a(14) to a(16) we have a run of three such terms: 6849, 9770, 17675. A yet longer such run is from a(23) to a(30): 2029875, 4059746, 7037655, 17594075, 50850483, 68589598, 186888243, 373659254.
%C A189760 Applying A327968 to these terms yields: 0, 0, 1, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, ...
%C A189760 Question: Are there indefinitely long sequences of iterations of A003415 that end with steps ... -> p -> 1 -> 0, with p=5? Are there such sequences for any other prime p? Can we construct a such sequence that is guaranteed to be infinite? See the subtree depicted in A327975 and conjecture #8 in Ufnarovski and Ahlander paper.
%C A189760 (End)
%H A189760 Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%F A189760 Least k such that A099307(k) = n.
%F A189760 For all n >= 0, A256750(a(n)) = A327966(a(n)) = n, A327965(a(n)) = A003415(a(n)). - _Antti Karttunen_, Oct 02 2019
%t A189760 nn = 15; t = Table[0, {nn}]; n = 0; cnt = 0; While[cnt < nn, n++; k = 0; d = n; While[f = Transpose[FactorInteger[d]]; d > 1 && And @@ MapThread[Greater, f], k++; d = Plus @@ (d*f[[2]]/f[[1]])]; If[d == 1, k++; If[k <= nn && t[[k]] == 0, t[[k]] = n; cnt++]]]; Join[{0},t]
%Y A189760 Cf. A003415, A098699, A099307, A327965, A327968, A327975, A327977.
%Y A189760 Cf. A256750, A327966 (left inverses for this sequence).
%Y A189760 Subsequence of A048103. Differs from A327967 for the first time at n=19.
%K A189760 nonn,more
%O A189760 0,3
%A A189760 _T. D. Noe_, Apr 27 2011
%E A189760 a(26)-a(31) from _Donovan Johnson_, Apr 29 2011