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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).

%I A306567 #15 May 29 2024 12:15:02
%S A306567 9,99,27,99,96,99,63,99,81,91,99,195,94,295,93,291,113,189,171,992,
%T A306567 159,187,187,483,988,475,153,281,181,273,279,577,297,997,567,369,333,
%U A306567 363,351,994,219,465,357,663,459,461,423,192,441,965,399,999,437,126,551
%N 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).
%C A306567 For any n > 0, a(n) is well defined:
%C A306567 - the set of zeroless numbers (A052382) contains arbitrarily large gaps,
%C A306567 - for example, for any k > 0, the interval I_k = [10^k..(10^(k+1)-1)/9-1] if free of zeroless numbers,
%C A306567 - let i be such that #I_i > n,
%C A306567 - let b_n be defined by b_n(0) = 0, and for any j > 0, b_n(j) = noz(b_n(j-1) + n),
%C A306567 - as b_n starts below 10^i and cannot cross the gap constituted by I_i,
%C A306567 - b_n is bounded (and eventually periodic), QED.
%H A306567 Rémy Sigrist, <a href="/A306567/b306567.txt">Table of n, a(n) for n = 1..10000</a>
%H A306567 Rémy Sigrist, <a href="/A306567/a306567.gp.txt">PARI program for A306567</a>
%F A306567 Empirically, for any k >= 0:
%F A306567 - a(    10^k) =  9 * 10^k +     (10^k-1)/9,
%F A306567 - a(2 * 10^k) = 99 * 10^k + 2 * (10^k-1)/9,
%F A306567 - a(3 * 10^k) = 27 * 10^k + 3 * (10^k-1)/9,
%F A306567 - a(4 * 10^k) = 99 * 10^k + 4 * (10^k-1)/9,
%F A306567 - a(5 * 10^k) = 96 * 10^k + 5 * (10^k-1)/9,
%F A306567 - a(6 * 10^k) = 99 * 10^k + 6 * (10^k-1)/9,
%F A306567 - a(7 * 10^k) = 63 * 10^k + 7 * (10^k-1)/9,
%F A306567 - a(8 * 10^k) = 99 * 10^k + 8 * (10^k-1)/9,
%F A306567 - a(9 * 10^k) = 81 * 10^k + 9 * (10^k-1)/9.
%e A306567 For n = 1:
%e A306567 - noz(0 + 1) = 1,
%e A306567 - noz(1 + 1) = 2,
%e A306567 - noz(2 + 1) = 3,
%e A306567   ...
%e A306567 - noz(7 + 1) = 8,
%e A306567 - noz(8 + 1) = 9,
%e A306567 - noz(9 + 1) = noz(10) = 1,
%e A306567 - hence a(1) = 9.
%o A306567 (PARI) \\ See Links section.
%Y A306567 See A306569 for the multiplicative variant.
%Y A306567 Cf. A004719, A052382.
%K A306567 nonn,look,base
%O A306567 1,1
%A A306567 _Rémy Sigrist_, Feb 24 2019