A189760 Least nonnegative number whose n-th arithmetic derivative (A003415) is zero and lower derivatives are nonzero.

0, 1, 2, 6, 9, 14, 33, 62, 177, 414, 1155, 1719, 2625, 4018, 6849, 9770, 17675, 30206, 90609, 260343, 336006, 757995, 1290874, 2029875, 4059746, 7037655, 17594075, 50850483, 68589598, 186888243, 373659254, 1884639669

Offset: 0

Author

T. D. Noe, Apr 27 2011

Keywords

• nonn
• more

Comments

a(32) <= 9519378185. - Donovan Johnson, Apr 30 2011

From Antti Karttunen, Oct 02 2019: (Start)

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.

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, ...

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.

(End)

Links

• Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Crossrefs

Cf. A003415, A098699, A099307, A327965, A327968, A327975, A327977.

Cf. A256750, A327966 (left inverses for this sequence).

Subsequence of A048103. Differs from A327967 for the first time at n=19.

Programs

Mathematica

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]

Formula

Least k such that A099307(k) = n.

For all n >= 0, A256750(a(n)) = A327966(a(n)) = n, A327965(a(n)) = A003415(a(n)). - Antti Karttunen, Oct 02 2019

Extensions

a(26)-a(31) from Donovan Johnson, Apr 29 2011