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.

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, 72, 99, 37, 12, 69, 64, 87, 12, 289, 27, 54, 82, 39, 42, 19, 6, 57, 37, 27, 54, 82, 12, 69, 64, 69, 12, 64, 27, 27, 82, 12, 87, 289, 69, 39, 289, 72, 99, 64, 57, 24

Offset: 0

Author

Rémy Sigrist, Nov 11 2018

Keywords

• nonn
• base

Comments

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.

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.

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.

Examples

			For n = 16:
- noz(15 + 16) = noz(31) = 31,
- noz(14 + 31) = noz(45) = 45,
- noz(13 + 45) = noz(58) = 58,
- noz(12 + 58) = noz(70) = 7,
- noz(11 + 7) = noz(18) = 18,
- noz(10 + 18) = noz(28) = 28,
- noz(9 + 28) = noz(37) = 37,
- noz(8 + 37) = noz(45) = 45,
- noz(7 + 45) = noz(52) = 52,
- noz(6 + 52) = noz(58) = 58,
- noz(5 + 58) = noz(63) = 63,
- noz(4 + 63) = noz(67) = 67,
- noz(3 + 67) = noz(70) = 7,
- noz(2 + 7) = noz(9) = 9,
- noz(1 + 9) = noz(10) = 1,
- hence a(16) = 1.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000 (corrected by _Paolo Xausa_, Apr 17 2024)

Crossrefs

Cf. A000217, A004719, A243658.

Programs

Mathematica

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A321480[n_] := Block[{k = n}, Nest[noz[--k + #] &, n, Max[0, n-1]]];
Array[A321480,100,0] (* Paolo Xausa, Apr 17 2024 *)

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

Extensions

a(10), a(20), a(30), a(40), a(50) and a(60) corrected by Paolo Xausa, Apr 17 2024