A022342 -id:A022342 - cp's OEIS frontend

A380079 Start with a list of the positive integers L in increasing order. Then, at turn n>=1, element n jumps from its current position, m, to position m+n. Then a(n) = L(m+1).

2, 1, 2, 5, 3, 7, 4, 5, 10, 6, 7, 13, 8, 15, 9, 10, 18, 11, 20, 12, 13, 23, 14, 15, 26, 16, 28, 17, 18, 31, 19, 20, 34, 21, 36, 22, 23, 39, 24, 41, 25, 26, 44, 27, 28, 47, 29, 49, 30, 31, 52, 32, 54, 33, 34, 57, 35, 36, 60, 37, 62, 38, 39, 65, 40, 41, 68, 42, 70, 43, 44, 73, 45, 75

Offset: 1

Ali Sada and David Nacin, Jan 11 2025

nonn

It appears that a(n)=n+1 exactly when n belongs to A003622.  For all other positions, fill in the numbers 1,2,3,4, ... in increasing order. A proof should be possible using the theorem-prover Walnut, and is underway. - Jeffrey Shallit, Jan 12 2025
After infinitely many moves, the natural numbers are transformed back into their original order. This is because all numbers larger than n must jump over n, as they have a larger jump by definition. Therefore, n must return to its original place in the list. It appears that n returns to its place at A022342(n-1).
There seems to be an infinite number of ordered pairs of the form (k, k+1) such as (1,2), (4,5), (6,7), (9,10), (12,13), .. etc., and the first term in each pair is A003622.
Also, it seems that if we take the last number in the jump (i.e. L(m+n-1), we get natural numbers excluding the terms of A003622, which seems to be A022342.
The last appearance of k also seems to be A022342.
The graph has two "rays", one with slope ~ 1 on which lies about every third term, and one with slope 0.618 on which lie the other terms. - M. F. Hasler, May 09 2025

			We start with 1,2,3,4,5,6,7,8,9,10... First, 1 jumps over 2, so, a(1) = 2. In the second turn, 2 jumps over 1 and 3, so, a(2) =1. In the third turn, 3 jumps over 2,4, and 5, so a(3) = 2. And so on.

M. F. Hasler, Table of n, a(n) for n = 1..500, May 08 2025
Jeffrey Shallit, The Hurt-Sada Array and Zeckendorf Representations, arXiv:2501.08823 [math.NT], 2025. See p. 2.

Cf. A003622, A368050.

M = list()
def moves(n, L):
    if n == 0:
        return L
    return move(n, moves(n-1,L))
def move(n, L):
    pos = L.index(n)
    M.append(L[pos+1])
    return L[:pos]+L[pos+1:pos+n+1]+[n]+L[pos+n+1:]
def seq(n):
    return moves(n, list(range(1,5*n)))
seq(74)
print(M) # David Nacin via Seqfan, Jan 10 2025

def aupton(nn):
    L, out = list(range(1, 2*nn+2)), []
    for n in range(1, nn+1):
        m = L.index(n)
        out.append(L[m+1])
        L.insert(m+n+1, n)
        L.pop(m)
    return out
print(aupton(74)) # Michael S. Branicky, Jan 13 2025

From Jeffrey Shallit, Jan 13 2025: (Start)
If n belongs to A003622, then a(n) = n+1.  Otherwise, a(A022342(n)) = n-1 for n >= 1. This can be proved with the Walnut theorem prover.
Alternatively, let g = (1+sqrt(5))/2, and let p be the fractional part of (n+1)*g. If p < 2-g, then a(n) = n+1. If p > 2-g, then a(n) = floor((n+1)/g). (End)

A047488 Numbers that are congruent to {0, 2, 3, 5, 7} mod 8.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 43, 45, 47, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 75, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 103

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Different from A022342.

[n : n in [0..150] | n mod 8 in [0, 2, 3, 5, 7]]; // Wesley Ivan Hurt, Jul 31 2016

A047488:=n->8*floor(n/5)+[(0, 2, 3, 5, 7)][(n mod 5)+1]: seq(A047488(n), n=0..100); # Wesley Ivan Hurt, Jul 31 2016

Select[Range[0,150], MemberQ[{0, 2, 3, 5, 7}, Mod[#,8]]&] (* Harvey P. Dale, Mar 20 2011 *)

G.f.: x^2*(2+x+2*x^2+2*x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)). [Colin Barker, May 14 2012]
From Wesley Ivan Hurt, Jul 31 2016: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6, a(n) = a(n-5) + 8 for n > 5.
a(n) = (40*n - 35 - 2*(n mod 5) - 2*((n+1) mod 5) + 3*((n+2) mod 5) - 2*((n+3) mod 5) + 3*((n+4) mod 5))/25.
a(5k) = 8k-1, a(5k-1) = 8k-3, a(5k-2) = 8k-5, a(5k-3) = 8k-6, a(5k-4) = 8k-8. (End)

A295282 a(n) > n is chosen to minimize the difference between ratios a(n):n and n:(a(n) - n), so that they are matching approximations to the golden ratio.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 99, 100, 102, 104, 105, 107, 108

Offset: 1

Peter Munn, Nov 19 2017

nonn

The difference between the matching ratios is evaluated by dividing the larger by the smaller.
Take a rectangle A with sides n and n+m; remove a square of side n from one end to form rectangle B with sides n and m; scale B in the ratio (n+m):n to form rectangle C with a side n+m. Place A and C alongside, with edges of length n+m coinciding, to form rectangle D. For n > 0, let m_n be the m that has the coincident edges dividing D in nearest to equal proportions, then a(n) = n + m_n.
Compared with other neighboring values of n, the resulting proportions can be made most nearly equal when n is a Fibonacci number F(k) = A000045(k), k > 1, in which case a(n) is F(k+1). In contrast, if 2n is a Fibonacci number F(k), then a relatively good choice for rectangle A's longer side would be F(k+1)/2, except that F(k+1) is odd when F(k) is even, so F(k+1)/2 is halfway between integers.
a(n) is usually the same as A007067(n), but when 2n is a Fibonacci number, they sometimes differ. The first differences are a(4) = 7 = A007067(4) + 1 and a(72) = 117 = A007067(72) + 1. The author expects a(n) to differ from A007067(n) if and only if n is in A060645. The terms of A060645 are half the value of alternate even Fibonacci numbers.
More specifically, for k > 0: F(3k) is an even Fibonacci number, F(3k+1) is odd and a(F(3k)/2) = F(3k+1)/2 + 1/2; whereas A007067(F(6k+3)/2) = F(6k+4)/2 + 1/2, but A007067(F(6k)/2) = F(6k+1)/2 - 1/2.

			The matching ratios and the differences between them begin:
   2:1          1:1         2.0
   3:2          2:1         1.3333...
   5:3          3:2         1.1111...
   7:4          4:3         1.3125
   8:5          5:3         1.0416...
  10:6          6:4         1.1111...
  11:7          7:4         1.1136...
  13:8          8:5         1.015625
  15:9          9:6         1.1111...
  16:10        10:6         1.0416...
  18:11        11:7         1.0413...
  19:12        12:7         1.0827...
  21:13        13:8         1.0059...
  23:14        14:9         1.0561...
  24:15        15:9         1.0416...
  26:16        16:10        1.015625
  28:17        17:11        1.0657...
  29:18        18:11        1.0156...
  31:19        19:12        1.0304...
  32:20        20:12        1.0416...
  34:21        21:13        1.0022...
  ...
For n = 4:
if a(4) = 5, the matching ratios would be a(4):4 = 5:4 and 4:(a(4)-4) = 4:1, with the difference between them (larger divided by smaller) = (4/1) / (5/4) = 16/5 = 3.2;
if a(4) = 6, ratios would be 6:4 and 4:2, with difference = (4/2) / (6/4) = 16/12 = 1.333...;
if a(4) = 7, ratios would be 7:4 and 4:3, with difference = (7/4) / (4/3) = 21/16 = 1.3125;
if a(4) = 8, ratios would be 8:4 and 4:4, with difference = (8/4) / (4/4) = 32/16 = 2.0.
Any larger value for a(4) would give a difference between the ratios that exceeded 2.0, so a(4) = 7, as this achieves the minimum difference.
This example translates as follows into the geometry described early in the comments:
                      n            4           4
                      m            2           3
                     n+m           6           7
Rectangle A       n X (n+m)      4 X 6       4 X 7
Rectangle B         m X n        2 X 4       3 X 4
Scaling ratio      n:(n+m)        4:6         4:7
m scaled up       m*(n+m)/n      2*6/4       3*7/4
= side of C           l            3          5.25
Rectangle C       l X (n+m)      3 X 6      5.25 X 7
Rectangle D     (n+l) X (n+m)    7 X 6      9.25 X 7
proportion C/D     l/(n+l)        3/7      5.25/9.25
- as decimal                    0.4285...   0.5675...
- its difference from 0.5       0.0714...   0.0675...

A001622 gives the value of the golden ratio.

Cf. A000045, A007067, A022342, A060645.

a(n) = (m+n) > n so as to minimize (max((m+n)/n, n/m) / min((m+n)/n, n/m)).

a(n+1) = a(n) + 2 - floor((a(n)+2) * (a(n)+1-n) * (a(n)+1) * (a(n)-n) / (n+1)^4), with a(0) = 0 for the purpose of this calculation.

A357316 A distension of the Wythoff array by inclusion of intermediate rows. Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals. If S is the set such that Sum_{i in S} F_i is the Zeckendorf representation of n then A(n,k) = Sum_{i in S} F_{i+k-2}.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 2, 1, 0, 3, 3, 3, 3, 2, 0, 5, 5, 5, 4, 3, 2, 0, 8, 8, 8, 7, 5, 4, 3, 0, 13, 13, 13, 11, 8, 6, 4, 3, 0, 21, 21, 21, 18, 13, 10, 7, 5, 3, 0, 34, 34, 34, 29, 21, 16, 11, 8, 6, 4, 0, 55, 55, 55, 47, 34, 26, 18, 13, 9, 6, 4

Offset: 0

Peter Munn, Sep 23 2022

Note the Zeckendorf representation of 0 is taken to be the empty sum.
The Wythoff array A035513 is the subtable formed by rows 3, 11, 16, 24, 32, ... (A035337). If, instead, we use rows 2, 7, 10, 15, 20, ... (A035336) or 1, 4, 6, 9, 12, ... (A003622), we get the Wythoff array extended by 1 column (A287869) or 2 columns (A287870) respectively.
Similarly, using A035338 truncates by 1 column; and in general if S_k is column k of the Wythoff array then the rows here numbered by S_k form an array A_k that starts with column k-2 of the Wythoff array. (A_0 and A_1 are the 2 extended arrays mentioned above.) As every positive integer occurs exactly once in the Wythoff array, every row except row 0 of A(.,.) is a row of exactly one such A_k.
Columns 4 onwards match certain columns of the multiplication table for Knuth's Fibonacci (or circle) product (extended variant - see A135090 and formula below).
For k > 0, the first row to contain k is A348853(k).

			Example for n = 4, k = 3. The Zeckendorf representation of 4 is F_4 + F_2 = 3 + 1. So the values of i in the sums in the definition are 4 and 2; hence A(4,3) = Sum_{i = 2,4} F_{i+k-2} = F_{4+3-2} + F_{2+3-2} = F_5 + F_3 = 5 + 2 = 7.
Square array A(n,k) begins:
   n\k| 0   1    2    3    4    5    6
  ----+--------------------------------
   0  | 0   0    0    0    0    0    0  ...
   1* | 0   1    1    2    3    5    8  ...
   2  | 1   1    2    3    5    8   13  ...
   3  | 1   2    3    5    8   13   21  ...
   4* | 1   3    4    7   11   18   29  ...
   5  | 2   3    5    8   13   21   34  ...
   6* | 2   4    6   10   16   26   42  ...
   7  | 3   4    7   11   18   29   47  ...
   8  | 3   5    8   13   21   34   55  ...
   9* | 3   6    9   15   24   39   63  ...
  10  | 4   6   10   16   26   42   68  ...
  11  | 4   7   11   18   29   47   76  ...
  12* | 4   8   12   20   32   52   84  ...
  ...
The asterisked rows form the start of the extended Wythoff array (A287870).

Encyclopedia of Mathematics, Zeckendorf representation

Columns, some differing initially: A005206 (1), A022342 (3), A026274 (4), A101345 (5), A101642 (6).
Rows: A000045 (1), A000204 (4).
Related to subtable A287870 as A130128 (as a square) is to A054582.
Other subtables: A035513, A287869.
See the comments for the relationship to A003622, A035336, A035337, A035338, A348853.
See the formula section for the relationship to A003714, A022342, A135090, A356874.

A5206(m) = if(m>0,m-A5206(A5206(m-1)),0)
A(n,k) = if(k==2,n, if(k==1,A5206(n), if(k==0,n-A5206(n), A(n,k-2)+A(n,k-1)))) \\ simple encoding of formulas, not efficient

For n >= 0, k >= 0 unless stated otherwise:
A(n,k) = A356874(floor(A003714(n)*2^(k-1))).
A(n,1) = A005206(n).
A(n,2) = n.
A(n,k+2) = A(n,k) + A(n,k+1).
A(A022342(n+1),k) = A(n,k+1).
For k >= 4, A(n,k) = A135090(n,A000045(k-2)).

A372302 Numbers k for which the Zeckendorf representation A014417(k) ends with "1001".

Original entry on oeis.org

6, 19, 27, 40, 53, 61, 74, 82, 95, 108, 116, 129, 142, 150, 163, 171, 184, 197, 205, 218, 226, 239, 252, 260, 273, 286, 294, 307, 315, 328, 341, 349, 362, 375, 383, 396, 404, 417, 430, 438, 451, 459, 472, 485, 493, 506, 519, 527, 540, 548, 561, 574, 582, 595, 603

Offset: 1

A.H.M. Smeets, Apr 25 2024

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A000045, A005614, A014417.
Tree of Zeckendorf subsequences of positive integers partitioned by their suffix part S (except initial term or offset in some cases). $ is the empty string. length(S) =
0        1        2        3        4        5        6       7
----------------------------------------------------------------------
$:       0:       00:      000:     0000:    00000:   000000:
A000027  A022342  A026274  A101345  A101642  notOEIS  notOEIS
                                                      100000: 0100000:
                                                      A035340  <------
                                             10000:
                                             A035339
                                    1000:    01000:
                                    A035338  <------
                  10:      010:     0010:
                  A035336  <------  A134861
                                    1010:    01010:
                                    A134863  <------
                           100:     0100:
                           A035337  <------
         1:       01:      001:     0001:
         A003622  <------  A134859  A151915
                                    1001:    01001:
                                    A372302  <------
                           101:     0101:
                           A134860  <------
Suffixes 10^n, where ^ means n times repeated concatenation, are the (n+1)-th columns in the Wythoff array A083412 and A035513 (n >= 0).

Equals {A134859}\{A151915}.
a(n) = A134863(n) - 1 = A035338(n) + 1.
a(n) = a(n-1) + 8 + 5*A005614(n-2) = a(n-1) + F(6) + F(5)*A005614(n-2), n > 1, where F(k) is the k-th Fibonacci number (A000045).

A083162 a(n) is the smallest unused proper divisor or proper multiple of n such that a(n)/n != a(m)/m for all m < n.

Original entry on oeis.org

2, 1, 9, 16, 25, 36, 49, 64, 3, 90, 110, 132, 156, 182, 210, 4, 255, 288, 323, 360, 399, 440, 483, 528, 5, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 6, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 7, 2250, 2346, 2444, 2544

Offset: 1

Amarnath Murthy, Apr 25 2003

nonn

Previous name: a(n) is either a multiple or a divisor of n but not equal to n, such that a(n)/n = a(m)/m implies m = n and n/a(n)= m/a(m) also implies n = m. Also a(m) = a(n) if and only if m = n.

Equivalently, a(n) is the smallest integer k other than n that is a divisor or multiple of n such that k/n != a(m)/m for all m < n.

			a(3) = 9, a(3)/3 = 3 hence for no other m > 3, a(m) = 3m.
a(1000) = 1000*(1000-max{k: b(k) < 1000} + 1) = 1000*(1000-29+1) = 972000.

A022342 gives the sequence analogous to b(n) if we replace the multiplications in the definition by additions.

a(a(n)) = n; a(n) = (b(k)/(k + 1)) if n = b(k) for some k and a(n) = n*(n - max{k: b(k) < n} + 1) otherwise, where b(k) is the k-th number at which a(n) < n. (Equivalently, b(k) is the unique x for which a(x)/x = (k+1).) - Carl B. Carlson (carlsonc(AT)stolaf.edu), Jan 09 2005

More terms from Carl B. Carlson (carlsonc(AT)stolaf.edu), Jan 09 2005

Edited and more terms added, Jon E. Schoenfield, Feb 21 2021

A352583 a(n) is the value of the cell in the Wythoff array that lies in the next row and same column as the cell containing n.

Original entry on oeis.org

4, 7, 11, 6, 18, 9, 10, 29, 12, 15, 16, 14, 47, 17, 20, 24, 19, 26, 22, 23, 76, 25, 28, 32, 27, 39, 30, 31, 42, 33, 36, 37, 35, 123, 38, 41, 45, 40, 52, 43, 44, 63, 46, 49, 50, 48, 68, 51, 54, 58, 53, 60, 56, 57, 199, 59, 62, 66, 61, 73, 64, 65, 84, 67, 70, 71, 69, 102, 72, 75

Offset: 1

Michel Marcus, Mar 21 2022

From Kevin Ryde, Jun 05 2022: (Start)
a(n) is n with the "odd" part (A348853) of its Zeckendorf representation increased to the next greater "odd" number.
This increase is Zeckendorf digits +10 or +100 at the odd part, according to whether the final digits there are ..101 or ..001, respectively.
A354321(n) is the first of those three digits so that a(n) = n + Fibonacci(A035612(n) + 3 - A354321(n)).
(End)

			The Wythoff array (A035513 or A083412) begins:
   1    2    3    5    8 ...
   4    7   11   18   29 ...
   6   10   16   26   42 ...
   ...
so a(1) = 4, a(2) = 7, a(3) = 11, a(4) = 6, ...

Cf. A035513 and A083412 (Wythoff array), A003603 (row number), A035612 (column number).
Cf. A348853 (odd part), A354321 (above 01), A000045 (Fibonacci numbers).
Cf. A022342 (same row, next column).
Cf. A349102 (binary increase odd).

T(n,k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
cell(n) = for (r=1, oo, for (c=1, oo, if (T(r,c) == n, return([r, c])); if (T(r,c) > n, break););); \\ see A003603 and A035612
a(n) = {my(pos = cell(n)); T(pos[1]+1, pos[2]);}

{ my(phi=quadgen(5),s=phi-1,c=2*phi-3);
a(n) = my(t=n,k=3,r);
  until(r

cp's OEIS Frontend

A380079 Start with a list of the positive integers L in increasing order. Then, at turn n>=1, element n jumps from its current position, m, to position m+n. Then a(n) = L(m+1).

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A047488 Numbers that are congruent to {0, 2, 3, 5, 7} mod 8.

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A295282 a(n) > n is chosen to minimize the difference between ratios a(n):n and n:(a(n) - n), so that they are matching approximations to the golden ratio.

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A357316 A distension of the Wythoff array by inclusion of intermediate rows. Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals. If S is the set such that Sum_{i in S} F_i is the Zeckendorf representation of n then A(n,k) = Sum_{i in S} F_{i+k-2}.

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A372302 Numbers k for which the Zeckendorf representation A014417(k) ends with "1001".

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A083162 a(n) is the smallest unused proper divisor or proper multiple of n such that a(n)/n != a(m)/m for all m < n.

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A352583 a(n) is the value of the cell in the Wythoff array that lies in the next row and same column as the cell containing n.

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