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A321480 Zeroless analog of triangular numbers (version 2): a(0) = 0, and for any n > 0, a(n) = noz(1 + noz(2 + ... + noz((n-1) + n))), where noz(n) = A004719(n) omits the zeros from n.

%I A321480 #27 Apr 21 2024 11:39:35
%S A321480 0,1,3,6,1,15,3,28,9,18,19,39,6,28,15,12,1,9,99,37,39,177,64,69,39,19,
%T A321480 72,99,37,12,69,64,87,12,289,27,54,82,39,42,19,6,57,37,27,54,82,12,69,
%U A321480 64,69,12,64,27,27,82,12,87,289,69,39,289,72,99,64,57,24
%N A321480 Zeroless analog of triangular numbers (version 2): a(0) = 0, and for any n > 0, a(n) = noz(1 + noz(2 + ... + noz((n-1) + n))), where noz(n) = A004719(n) omits the zeros from n.
%C A321480 This sequence is a variant of A243658 where the additions are carried in the opposite order; as (i, j) -> noz(i + j) is not associative in general we obtain another sequence.
%C A321480 This sequence is conjectured to be bounded. This could be explained by the fact that the zeros appearing in the last steps of the calculation (when adding small values) erode the number of digits of the intermediate sums.
%C A321480 The distinct values among the first 1000000 terms are: 0, 1, 3, 6, 9, 12, 15, 18, 19, 24, 27, 28, 37, 39, 42, 54, 57, 64, 69, 72, 82, 84, 87, 99, 177, 289.
%H A321480 Rémy Sigrist, <a href="/A321480/b321480.txt">Table of n, a(n) for n = 0..10000</a> (corrected by _Paolo Xausa_, Apr 17 2024)
%e A321480 For n = 16:
%e A321480 - noz(15 + 16) = noz(31) = 31,
%e A321480 - noz(14 + 31) = noz(45) = 45,
%e A321480 - noz(13 + 45) = noz(58) = 58,
%e A321480 - noz(12 + 58) = noz(70) = 7,
%e A321480 - noz(11 + 7) = noz(18) = 18,
%e A321480 - noz(10 + 18) = noz(28) = 28,
%e A321480 - noz(9 + 28) = noz(37) = 37,
%e A321480 - noz(8 + 37) = noz(45) = 45,
%e A321480 - noz(7 + 45) = noz(52) = 52,
%e A321480 - noz(6 + 52) = noz(58) = 58,
%e A321480 - noz(5 + 58) = noz(63) = 63,
%e A321480 - noz(4 + 63) = noz(67) = 67,
%e A321480 - noz(3 + 67) = noz(70) = 7,
%e A321480 - noz(2 + 7) = noz(9) = 9,
%e A321480 - noz(1 + 9) = noz(10) = 1,
%e A321480 - hence a(16) = 1.
%t A321480 noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
%t A321480 A321480[n_] := Block[{k = n}, Nest[noz[--k + #] &, n, Max[0, n-1]]];
%t A321480 Array[A321480,100,0] (* _Paolo Xausa_, Apr 17 2024 *)
%o A321480 (PARI) a(n, base=10) = { my (t=n); forstep (k=n-1, 1, -1, t = fromdigits(select(sign, digits(t+k, base)), base)); t } \\ corrected by _Rémy Sigrist_, Apr 17 2024
%Y A321480 Cf. A000217, A004719, A243658.
%K A321480 nonn,base
%O A321480 0,3
%A A321480 _Rémy Sigrist_, Nov 11 2018
%E A321480 a(10), a(20), a(30), a(40), a(50) and a(60) corrected by _Paolo Xausa_, Apr 17 2024