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

Author

Antti Karttunen, Feb 11 2022

Keywords

• nonn
• look

Comments

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

Examples

			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.
		

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)

Crossrefs

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

Programs

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, ", ")));

Formula

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

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