A175402 a(n) is the number of iterations of {r -> (((D_1^D_2)^D_3)^...)^D_k, where D_k is the k-th decimal digit of r} needed to reach a one-digit number, starting at r = n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 4, 1, 1, 1, 3, 2, 3, 3, 3, 3, 2, 1, 1, 2, 3, 3, 2, 2, 2, 3, 3, 1, 1, 3, 2, 3, 3, 2, 3, 2, 2, 1, 1, 4, 4, 2, 3, 3, 3, 2, 2, 1, 1, 4, 4, 2, 2, 2, 3, 2, 2, 1, 1, 3, 4, 2, 3, 3, 2, 2, 2, 1, 1, 2, 3, 3, 2, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset

0,25

Comments

Conjecture: max(a(n)) = 4.

Assuming that A020665(2) = 86, A020665(3) = 68, A020665(5) = 58, and A020665(7) = 35, this conjecture is true, since in that case the largest power of a decimal digit that has no 0 is 7^35, and of those powers p none have a(p) > 3. The only n for which a(n) = 4 are those where one iteration goes to 6^2, 7^2, 6^3, 7^3, or 2^9.

Links

Table of n, a(n) for n=0..107.

Example

For n = 29: a(29) = 4 because for the number 29 there are 4 steps of defined iteration: {2^9 = 512}, {(5^1)^2 = 25}, {2^5 = 32}, {3^2 = 9}.

Prog

(PARI) iter(n)=my(v=eval(Vec(Str(n)))); v[1]^prod(i=2, #v, v[i])
a(n)=my(k=0); while(n>9, k++; n=iter(n)); k

Crossrefs

Cf. A175398, A175399, A175400, A175401, A175403, A175405.

Sequence in context: A285734 A051698 A046773 * A281726 A101037 A002199

Adjacent sequences: A175399 A175400 A175401 * A175403 A175404 A175405

Keyword

nonn,base

Author

Jaroslav Krizek, May 01 2010

Extensions

Corrected, extended, comment, and program from Charles R Greathouse IV, Aug 03 2010

Status

approved