A028897 -id:A028897 - cp's OEIS frontend

A249873 In decimal representation: replace each even positioned digit with the digit sum of its double.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36

Offset: 0

Reinhard Zumkeller, Nov 08 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to decimal expansion of n

Cf. A007953, A093017, A028897.

a249873 n = if n == 0 then 0 else 100*a249873 n' + 10*a007953 (2*t) + d
            where (n', td) = divMod n 100; (t, d) = divMod td 10

a(n) = my(d=Vecrev(digits(n))); forstep (i=2, #d, 2, d[i] = sumdigits(2*d[i])); fromdigits(Vecrev(d)); \\ Michel Marcus, Jan 23 2022

A322001 Digits of n interpreted in factorial base: a(Sum d_k10^k) = Sum d_kk!

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6

Offset: 0

M. F. Hasler, Nov 27 2018

More terms than usual are given to distinguish the sequence from A081594, A028897 and A244158, which agree up to a(99). The last two correspond to k! replaced by 2^k resp. Catalan(k).

This is a left inverse to A007623 (factorial base representation of n): A322001(A007623(n)) = n for all n >= 0. One could imagine variants which have a(n) = 0 or a(n) = -1 if n is not a term of A007623. Restricted to the range of A007623, it is also a right inverse to A007623, at least up to the 10 digit terms, beyond which A007623 becomes non-injective.

Cf. A007623 (right inverse), A081594, A028897, A244158.

a[n_] := Module[{d=Reverse@IntegerDigits[n]}, Sum[d[[i]]*i!, {i,1,Length[d]}]]; Array[a, 100, 0] (* Amiram Eldar, Nov 28 2018 *)

A322001(n)=sum(i=1,#n=Vecrev(digits(n)),n[i]*i!) \\ M. F. Hasler, Nov 27 2018

A083291 Triangular array read by rows: T(n,k) = k*floor(n/10) + n mod 10, 0<=k<=n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 0

Reinhard Zumkeller, Apr 23 2003

A010879(n)=T(n,0);
A076314(0)=T(0,0), A076314(n)=T(n,1) for n>0;
A028897(n)=T(n,n) for n<=1, A028897(n)=T(n,2) for n>1;
A028898(n)=T(n,n) for n<=2, A028898(n)=T(n,3) for n>2;
A028899(n)=T(n,n) for n<=3, A028899(n)=T(n,4) for n>3;
A028900(n)=T(n,n) for n<=4, A028900(n)=T(n,5) for n>4;
A028901(n)=T(n,n) for n<=5, A028901(n)=T(n,6) for n>5;
A028902(n)=T(n,n) for n<=6, A028902(n)=T(n,7) for n>6;
A028903(n)=T(n,n) for n<=7, A028903(n)=T(n,8) for n>7;
A028904(n)=T(n,n) for n<=8, A028904(n)=T(n,9) for n>8;
T(n,n) = n for n<=9, T(n,10) = n for n>9;
A083292(n) = T(n,n).

			From _Paolo Xausa_, May 22 2024: (Start)
Triangle begins:
   [0] 0;
   [1] 1, 1;
   [2] 2, 2, 2;
   [3] 3, 3, 3, 3;
   [4] 4, 4, 4, 4, 4;
   [5] 5, 5, 5, 5, 5, 5;
   [6] 6, 6, 6, 6, 6, 6, 6;
   [7] 7, 7, 7, 7, 7, 7, 7, 7;
   [8] 8, 8, 8, 8, 8, 8, 8, 8, 8;
   [9] 9, 9, 9, 9, 9, 9, 9, 9, 9, 9;
  [10] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
  ... (End)

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Table[k*Floor[n/10] + Mod[n, 10], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, May 22 2024 *)

Offset changed to 0 by Paolo Xausa, May 22 2024

A061196 If n = Sum_{i} c_i * 10^i then let b(n) = Sum_{i} c_i * (i+1). Order the integers by b(n) and then n.

Original entry on oeis.org

0, 1, 2, 10, 3, 11, 100, 4, 12, 20, 101, 1000, 5, 13, 21, 102, 110, 1001, 10000, 6, 14, 22, 30, 103, 111, 200, 1002, 1010, 10001, 100000, 7, 15, 23, 31, 104, 112, 120, 201, 1003, 1011, 1100, 10002, 10010, 100001, 1000000, 8, 16, 24, 32, 40, 105, 113, 121, 202

Offset: 0

Henry Bottomley, Apr 20 2001

This is in effect a listing of single-digit (nonnegative) solutions to b + 2c + 3d + 4e + ... = k.

The sequence can be considered as an irregular triangle listing partitions in which no part occurs more than 9 times. The row lengths are given by A261776. For example, in row 5 the value 102, corresponds to the partition 1+1+3 (= 2*1 + 0*2 + 1*3). - Andrew Howroyd, Apr 25 2023

			From _Andrew Howroyd_, Apr 25 2023: (Start)
The sequence as a triangle T(n,k) begins:
  0 | 0;
  1 | 1;
  2 | 2, 10;
  3 | 3, 11, 100;
  4 | 4, 12,  20, 101, 1000;
  5 | 5, 13,  21, 102,  110, 1001, 10000;
  6 | 6, 14,  22,  30,  103,  111,   200, 1002, 1010, 10001, 100000;
  ...
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..96
Rasa Smidtaite and Minvydas Ragulskis, Commentary: Multidimensional discrete chaotic maps, Front. Phys. (2022) Vol. 10, 1094240.

Cf. A000041, A028897, A261776.

With[{k = 7}, {{0}}~Join~Values@ PositionIndex[Array[Total@ MapIndexed[#1*First[#2] &, Reverse@ IntegerDigits[#]] &, 10^k]][[1 ;; k]]] // Flatten (* Michael De Vlieger, Dec 22 2022, solution only suitable for generating the data field *)

F(p)={my(v=vector(if(#p, p[#p], 1))); for(i=1, #p, v[p[i]]++); v}
row(n)={my(R=[F(p) | p<-partitions(n)]); vecsort([fromdigits(Vecrev(u)) | u<-R, vecmax(u)<=9])}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Apr 25 2023

For n < 10, a(A000070(n)) = n+1 and a(A026905(n)) = 10^(n-1).

A309616 a(n) is the number of ways to represent 2*n in the decibinary system.

Original entry on oeis.org

1, 2, 4, 6, 10, 13, 18, 22, 30, 36, 45, 52, 64, 72, 84, 93, 110, 122, 140, 154, 177, 192, 214, 230, 258, 277, 304, 324, 356, 376, 405, 426, 464, 490, 528, 557, 604, 634, 678, 710, 765, 802, 854, 892, 952, 989, 1042, 1080, 1146, 1190, 1253, 1300, 1374, 1420, 1486, 1533, 1612, 1664

Offset: 0

Jonas Hollm, Aug 10 2019

It appears that a(n) is the number of decibinary numbers that can be constructed to represent the decimal numbers 2n-2 and 2n-1. To make this more clear let's consider n = 5: a(5) = 10 means that there are 10 decibinary numbers that represent the decimal numbers 2*5 - 2 = 8 and 2*5 - 1 = 9.

Furthermore, a(n) is the number of k such that A028897(k)=2*n.

			a(1) = 1.
a(2) = a(2-1) + a(ceiling(2/2)) = a(1) + a(1) = 1 + 1 = 2.
a(3) = a(3-1) + a(ceiling(3/2)) = a(2) + a(2) = 2 + 2 = 4.
a(4) = a(4-1) + a(ceiling(4/2)) = a(3) + a(2) = 4 + 2 = 6.
a(5) = a(5-1) + a(ceiling(5/2)) = a(4) + a(3) = 6 + 4 = 10.
a(6) = a(6-1) + a(ceiling(6/2)) - a(ceiling((6-5)/2)) = a(5) + a(3) - a(1) = 10 + 4 - 1 = 13.
a(7) = a(7-1) + a(ceiling(7/2)) - a(ceiling((7-5)/2)) = a(6) + a(4) - a(1) = 13 + 6 - 1 = 18.
a(8) = a(8-1) + a(ceiling(8/2)) - a(ceiling((8-5)/2)) = a(7) + a(4) - a(2) = 18 + 6 - 2 = 22.
a(9) = a(9-1) + a(ceiling(9/2)) - a(ceiling((9-5)/2)) = a(8) + a(5) - a(2) = 22 + 10 - 2 = 30.
a(10) = a(10-1) + a(ceiling(10/2)) - a(ceiling((10-5)/2)) = a(9) + a(5) - a(3) = 30 + 10 - 4 = 36.

HackerRank, Decibinary Numbers.

Cf. A007728: superseeker found that the deltas of the sequence a(n+1) - a(n) match transformations of the original query.

Cf. A028897.

Nest[Append[#1, #1[[-1]] + #1[[Ceiling[#2/2] ]] - If[#2 > 5, #1[[Ceiling[(#2 - 5)/2] ]], 0 ]] & @@ {#, Length@ # + 1} &, {1}, 57] (* Michael De Vlieger, Sep 29 2019 *)

a(1) = 1. a(n) = a(n-1) + a(ceiling(n/2)) if 1 < n <= 5.
Conjecture: a(n) = a(n-1) + a(ceiling(n/2)) - a(ceiling((n-5)/2)) if n > 5.
I think this sequence is closely related to the 10th binary partition function. The only difference is that every second number is omitted. At the moment, the 10th binary partition function is not in the OEIS. However, my experiments strongly suggest that the 10th binary partition function would indeed look like 1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 13, 13, ...

Name corrected by Rémy Sigrist, Oct 15 2019

A322010 Inverse permutation to A322000.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 14, 20, 26, 36, 3, 5, 7, 11, 15, 21, 27, 37, 46, 59, 8, 12, 16, 22, 28, 38, 47, 60, 72, 90, 17, 23, 29, 39, 48, 61, 73, 91, 108, 130, 30, 40, 49, 62, 74, 92, 109, 131, 152, 182, 50, 63, 75, 93, 110, 132, 153, 183, 212, 248, 76, 94, 111, 133, 154, 184, 213, 249

Offset: 0

M. F. Hasler, Feb 19 2019

nonn

a(n) is the position of n in the list A322000 of "decibinary numbers", i.e., integers sorted according to their decibinary value A028897(n) = Sum d[i]*2^i, where d[i] are the decimal digits of n.
For 0 <= m <= 9, we have a(n) = A322003(n) = A000123(n-1), because 1..9 are the first few terms of A322000 where the decibinary value increases.
We see that a(10..19) = a(2..9)+1 concatenated with (46, 49). Then, a(20..29) = a(12..19)+1 concatenated with (72, 90). Then, a(30..39) = a(22..29)+1 concatenated with (108, 130), and so on. This yields an alternate way to compute the sequence.

Cf. A322000, A028897, A322003, A072170(.,10), A000123.

vec_A322010=vecsort(A,,1)[1..vecmin(setminus([1..#A],Set(A)))-1] \\ Assumes the vector A = A322000(1..N) has been computed for some N. Exclude initial 0's to have correct (1-based) indices of the vectors.

cp's OEIS Frontend

A249873 In decimal representation: replace each even positioned digit with the digit sum of its double.

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A322001 Digits of n interpreted in factorial base: a(Sum d_k*10^k) = Sum d_k*k!

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A083291 Triangular array read by rows: T(n,k) = k*floor(n/10) + n mod 10, 0<=k<=n.

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A061196 If n = Sum_{i} c_i * 10^i then let b(n) = Sum_{i} c_i * (i+1). Order the integers by b(n) and then n.

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A309616 a(n) is the number of ways to represent 2*n in the decibinary system.

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A322010 Inverse permutation to A322000.

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A322001 Digits of n interpreted in factorial base: a(Sum d_k10^k) = Sum d_kk!