A248025 -id:A248025 - cp's OEIS frontend

A247879 Inverse permutation to A248025.

1, 4, 7, 10, 13, 16, 19, 22, 25, 2, 3, 28, 37, 40, 55, 64, 67, 82, 91, 5, 6, 33, 43, 53, 57, 70, 77, 90, 94, 8, 9, 29, 46, 51, 59, 73, 81, 89, 97, 11, 12, 34, 38, 49, 61, 66, 76, 88, 100, 14, 15, 30, 41, 44, 63, 69, 80, 87, 103, 17, 18, 35, 47, 54, 56, 72

Offset: 1

Reinhard Zumkeller, Sep 30 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

import Data.List (elemIndex); import Data.Maybe (fromJust)
a247879 = (+ 1) . fromJust . (`elemIndex` a248025_list)

A330521 Lexicographically earliest sequence of distinct positive terms such that the digitsum of a(n) ends a(n+1).

Original entry on oeis.org

1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 10, 21, 23, 25, 27, 29, 111, 33, 26, 28, 110, 22, 24, 36, 39, 112, 34, 37, 210, 43, 47, 211, 44, 38, 311, 35, 48, 212, 45, 49, 113, 55, 310, 54, 59, 114, 46, 410, 65, 411, 56, 511, 57, 312, 66, 412, 67, 213, 76, 313, 77, 214, 87, 115, 97, 116, 58, 413, 68, 314

Offset: 1

Carole Dubois and Eric Angelini, Mar 01 2020

This is conjectured to be a permutation of the integers > 0.

			a(1) = 1 has digitsum 1, and this 1 ends a(2) = 11;
a(2) = 11 has digitsum 2 and this 2 ends a(3) = 2;
a(3) = 2 has digitsum 2 and this 2 ends a(4) = 12;
a(4) = 12 has digitsum 3 and this 3 ends a(5) = 3;
a(5) = 3 has digitsum 3 and this 3 ends a(6) = 13;
...
a(18) = 19 has digitsum 10 and this 10 ends a(19) = 10;
a(19) = 10 has digitsum 1 and this 1 ends a(20) = 21 (as 1 and 11 are already in the sequence); etc.

Carole Dubois, Table of n, a(n) for n = 1..5000

Cf. A248025.

A334737 Lexicographically earliest sequence of distinct positive integers such that the digital root of a(n+1) divides a(n).

Original entry on oeis.org

1, 10, 2, 11, 19, 28, 4, 13, 37, 46, 20, 5, 14, 7, 16, 8, 17, 55, 23, 64, 22, 29, 73, 82, 38, 47, 91, 25, 32, 26, 56, 31, 100, 40, 35, 34, 65, 41, 109, 118, 74, 83, 127, 136, 44, 49, 43, 145, 50, 59, 154, 52, 58, 92, 67, 163, 172, 76, 85, 68, 94, 101, 181, 190, 77, 61, 199

Offset: 1

Eric Angelini and Carole Dubois, May 09 2020

			a(1) = 1 is divisible by the digital root of 10 (which is 1 + 0 = 1);
a(2) = 10 is divisible by the dig. root of 2 (which is = 2);
a(3) = 2 is divisible by the dig. root of 11 (which is 1 + 1 = 2);
a(4) = 11 is divisible by the dig. root of 19 (which is 1 + 9 = 10 => 1 + 0 = 1);
a(5) = 19 is divisible by the dig. root of 28 (which is 2 + 8 = 10 => 1 + 0 = 1);
a(6) = 28 is divisible by the dig. root of 4 (which is = 4); etc.

Carole Dubois, Table of n, a(n) for n = 1..5005

Cf. A248025, A334837 (same idea, but digital sum instead of digital root in the Name section).

A338191 a(1) = 1, a(n) is the least m not already in a(n) such that m mod 10 = decimal digital root of a(n - 1).

Original entry on oeis.org

1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 21, 23, 25, 27, 29, 22, 24, 26, 28, 31, 34, 37, 41, 35, 38, 32, 45, 39, 33, 36, 49, 44, 48, 43, 47, 42, 46, 51, 56, 52, 57, 53, 58, 54, 59, 55, 61, 67, 64, 71, 68, 65, 62, 78, 66, 63, 69, 76, 74, 72

Offset: 1

Michael De Vlieger, Oct 15 2020

Define d(n) as the decimal digital root of n, which is equivalent to n = r (mod 9), replacing the residue r = 0 with 9 in all cases of nonzero n.
m = 0 (mod 10) is prohibited as a consequence, therefore a(n) is not a permutation of the natural numbers, but contains all positive nonzero m indivisible by 10.
We may write the function d(n) instead as "->" for brevity, separating the least novel m from r with a colon. Therefore, a(2) is derived from a(1) = 1 thus: 1 -> 1: 11 (see Example).
Having found a(1)..a(81), we may generate a(81k + j) = 90k + a(j), since a(82) = 91 -> 1 and the next interval of 90 unused numbers are congruent to 0 < m < 90 (mod 9). By induction we see the sequence is infinite and contains all nonzero m (mod 90) that are indivisible by 10.
Graphing very many terms results in a line-like plot with slope 10/9. Compare the behavior and plot of this sequence to A248025, which applies d(a(n-1)) to the first digit of m rather than last.

			The sequence repeats 8 phases generally related to m mod 90 by decade.
Phase 1 containing a(n) with 1 <= n <= 18 and involving 1 <= m mod 90 <= 19, begins as follows: 1 -> 1: 11 -> 2: 2 -> 2: 12 -> 3: 3, etc., therefore we have {1, 11, 2, 12, 3, 13, ..., 19}, wherein we have each r twice in succession but incrementing r afterward.
Phase 2 containing a(n) with 19 <= n <= 27 and involving m mod 90 in the 20s, results from 19 -> 2, the third request for r = 2, so a(19) = 21. 21 -> 3: 23 -> 5: 25, etc. thus {21, 23, 25, 27, 29}, then 29 -> 2: 22 -> 4: 24, etc. thus {22, 24, 26, 28}.
Phase 3 contains a(n) with 28 <= n <= 36: 28 -> 1: 31 -> 4: 34 -> 7: 37 -> 1: 41 -> 5: 35 -> 8: 38 -> 2: 32 -> 5: 45 -> 9: 39 -> 3: 33 -> 6: 36. This exhausts m mod 90 in the thirties. Generally, phases 3 | p involve m mod 90 = 10*p + c*(p + 1), with 0 <= c <= 1.
Phase 4 contains a(n) with 37 <= n <= 45 and begins with {41, 45} already used. 36 -> 9: 49 -> 4: 44 -> 8: 48 -> 3: 43 -> 7: 47 -> 2: 42 -> 6: 46. This exhausts m mod 90 in the forties.
Phase 5 contains a(n) with 46 <= n <= 54: 46 -> 1: 51 -> 6: 56 -> 2: 52, etc., thus {51, 56, 52, 57, 53, 58, 54, 59, 55}, exhausting m mod 90 in the fifties.
Phase 6 contains a(n) with 55 <= n <= 65: 55 -> 1: 61 -> 7: 67 -> 4: 64 -> 1: 71 -> 8: 68 -> 5: 65 -> 2: 62 -> 8: 78 -> 6: 66 -> 3: 63 -> 9: 69. We have exhausted m mod 90 in the sixties.
Phase 7 contains a(n) with 66 <= n <= 72, begining with {71, 78} already used. 69 -> 6: 76, etc., thus {76, 74, 72, 79, 77, 75, 73}, exhausting m mod 90 in the seventies.
Phase 8 is the last phase, ending with a(81): 73 -> 1: 81 -> 9: 89, etc., thus {81, 89, 88, ..., 83, 82}.
Therefore we have generated a(1)..a(81) and may express a(n) for n > 81 via a(81k + j) = 90k + a(j).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of the first 81 terms, points colored per m mod 10.

Cf. A248025.

With[{s = Nest[Append[#, Block[{k = 1, r = Mod[#[[-1]], 9] + 9 Boole[Mod[#[[-1]], 9] == 0]}, While[Nand[FreeQ[#, k], Mod[k, 10] == r], k++]; k]] &, {1}, 9^2]}, Array[If[#2 == 0, 90 #1 - 8, 90 #1 + s[[#2]] ] & @@ QuotientRemainder[#, 81] &, 10^3]]

A340138 a(1)=0, a(n+1) = the least m not already in the sequence such that the decimal digital root r of a(n-1) is found somewhere among the decimal digits of m.

Original entry on oeis.org

0, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 21, 23, 25, 27, 29, 20, 22, 24, 26, 28, 31, 34, 37, 41, 35, 38, 32, 45, 39, 30, 33, 36, 49, 40, 42, 46, 51, 56, 52, 47, 62, 48, 43, 57, 53, 58, 44, 68, 50, 54, 59, 55, 61, 67, 64, 71, 78, 60

Offset: 1

Michael De Vlieger, Jan 21 2021

a(n) = 10^k occurs before n = 10^k for 0 < k <= 5.
Conjecture: permutation of the nonnegative numbers.
A decimal version of A339607, a permutation of the integers having to do with binary weight, but instead using digital root rather than digit sum. The plots of these sequences have similar features.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot (n, a(n)) for 1 <= n <= 10^3 with records in dark red and least unused numbers in dark blue, with records of a(n)-n in red, many labeled, and records of n-a(n) in blue, many labeled.
Michael De Vlieger, Logarithmic plot (n, a(n)-n) for 1 <= n <= 10^5 with records labeled in red, records of n-a(n) labeled in blue, and zeros plotted in black.

Cf. A248025, A338191, A339607.

Block[{a = {0, 10}, k, r}, Do[k = 1; r = # + 9 Boole[# == 0] &@ Mod[a[[-1]], 9]; While[Nand[FreeQ[a, k], ! FreeQ[IntegerDigits[k], r]], k++]; AppendTo[a, k], 66]; a]

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A247879 Inverse permutation to A248025.

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A330521 Lexicographically earliest sequence of distinct positive terms such that the digitsum of a(n) ends a(n+1).

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A334737 Lexicographically earliest sequence of distinct positive integers such that the digital root of a(n+1) divides a(n).

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A338191 a(1) = 1, a(n) is the least m not already in a(n) such that m mod 10 = decimal digital root of a(n - 1).

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A340138 a(1)=0, a(n+1) = the least m not already in the sequence such that the decimal digital root r of a(n-1) is found somewhere among the decimal digits of m.

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