A306567 a(n) is the largest value obtained by iterating x -> noz(x + n) starting from 0 (where noz(k) = A004719(k) omits the zeros from k).

9, 99, 27, 99, 96, 99, 63, 99, 81, 91, 99, 195, 94, 295, 93, 291, 113, 189, 171, 992, 159, 187, 187, 483, 988, 475, 153, 281, 181, 273, 279, 577, 297, 997, 567, 369, 333, 363, 351, 994, 219, 465, 357, 663, 459, 461, 423, 192, 441, 965, 399, 999, 437, 126, 551

Offset: 1

Author

Rémy Sigrist, Feb 24 2019

Keywords

• nonn
• look
• base

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.

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A306567

Crossrefs

See A306569 for the multiplicative variant.

Cf. A004719, A052382.

Programs

PARI

\\ See Links section.

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.