A351257 Least k such that the k-th arithmetic derivative of A351255(n) is zero.

1, 2, 2, 3, 4, 5, 2, 3, 3, 4, 5, 6, 5, 6, 2, 5, 4, 3, 4, 3, 3, 4, 5, 4, 4, 7, 12, 6, 7, 4, 4, 4, 4, 4, 4, 4, 5, 8, 8, 6, 5, 12, 5, 5, 5, 8, 10, 6, 6, 6, 7, 6, 7, 7, 8, 12, 8, 12, 2, 3, 6, 3, 3, 4, 4, 5, 4, 4, 4, 8, 5, 5, 6, 12, 6, 3, 3, 7, 3, 3, 10, 3, 4, 5, 5, 4, 4, 6, 4, 4, 4, 6, 5, 5, 6, 5, 9, 5, 6, 10, 7, 7, 7

Offset

1,2

Comments

a(n) is the number of iterations of the map x -> A003415(x) needed to reach zero, when starting from x = A351255(n).

Links

Antti Karttunen, Table of n, a(n) for n = 1..105368 (computed for all 19-smooth terms of A351255, and also for A276086(9699690) = 23)

Formula

a(n) = A099307(A351255(n)).

For all n, a(n) > A351256(n). [See A351258 for the differences].

Example

From A351255(27) = 2625 it takes 12 iterations of map x -> A003415(x) to reach zero, as 2625 -> 2825 -> 1155 -> 886 -> 445 -> 94 -> 49 -> 14 -> 9 -> 6 -> 5 -> 1 -> 0, therefore a(27) = 12.

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));
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n, s, 0); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0, print1(c, ", ")));

Crossrefs

Cf. A003415, A099307, A276086, A351255, A351256, A351258, A351259, A351261.

Sequence in context: A036809 A111263 A357378 * A286626 A328628 A328629

Adjacent sequences: A351254 A351255 A351256 * A351258 A351259 A351260

Keyword

nonn,look

Author

Antti Karttunen, Feb 11 2022

Status

approved