OFFSET
1,1
COMMENTS
For any n > 0, a(n) is well defined:
- the set of zeroless numbers (A052382) contains arbitrarily large gaps,
- for example, for any k > 0, the interval I_k = [10^k..(10^(k+1)-1)/9-1] if free of zeroless numbers,
- let i be such that #I_i > n,
- let b_n be defined by b_n(0) = 0, and for any j > 0, b_n(j) = noz(b_n(j-1) + n),
- as b_n starts below 10^i and cannot cross the gap constituted by I_i,
- b_n is bounded (and eventually periodic), QED.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A306567
FORMULA
Empirically, for any k >= 0:
- a( 10^k) = 9 * 10^k + (10^k-1)/9,
- a(2 * 10^k) = 99 * 10^k + 2 * (10^k-1)/9,
- a(3 * 10^k) = 27 * 10^k + 3 * (10^k-1)/9,
- a(4 * 10^k) = 99 * 10^k + 4 * (10^k-1)/9,
- a(5 * 10^k) = 96 * 10^k + 5 * (10^k-1)/9,
- a(6 * 10^k) = 99 * 10^k + 6 * (10^k-1)/9,
- a(7 * 10^k) = 63 * 10^k + 7 * (10^k-1)/9,
- a(8 * 10^k) = 99 * 10^k + 8 * (10^k-1)/9,
- a(9 * 10^k) = 81 * 10^k + 9 * (10^k-1)/9.
EXAMPLE
For n = 1:
- noz(0 + 1) = 1,
- noz(1 + 1) = 2,
- noz(2 + 1) = 3,
...
- noz(7 + 1) = 8,
- noz(8 + 1) = 9,
- noz(9 + 1) = noz(10) = 1,
- hence a(1) = 9.
PROG
(PARI) \\ See Links section.
CROSSREFS
KEYWORD
AUTHOR
Rémy Sigrist, Feb 24 2019
STATUS
approved