A020703 -id:A020703 - cp's OEIS frontend

A038722 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,2,3,4,... .

1, 3, 2, 6, 5, 4, 10, 9, 8, 7, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 28, 27, 26, 25, 24, 23, 22, 36, 35, 34, 33, 32, 31, 30, 29, 45, 44, 43, 42, 41, 40, 39, 38, 37, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 78, 77, 76

Offset: 1

N. J. A. Sloane, May 02 2000

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The rectangular array having A038722 as antidiagonals is the transpose of the rectangular array given by A000217. Column 1 of array A038722 is A000124 (central polygonal numbers). Array A038722 is the dispersion of the complement of A000124. - Clark Kimberling, Apr 05 2003
a(n) is the smallest number not yet in the sequence such that n + a(n) is one more than a square. - Franklin T. Adams-Watters, Apr 06 2009
From Hieronymus Fischer, Apr 30 2012: (Start)
A reordering of the natural numbers.
The sequence is self-inverse in that a(a(n)) = n.
Also: a(1) = 1, a(n) = m (where m is the least triangular number > a(k) for 1 <= k < n), if the minimal natural number not yet in the sequence is greater than a(n-1), otherwise a(n) = a(n-1)-1. (End)

			The rectangular array view is
   1    2    4    7   11   16   22   29   37   46
   3    5    8   12   17   23   30   38   47   57
   6    9   13   18   24   31   39   48   58   69
  10   14   19   25   32   40   49   59   70   82
  15   20   26   33   41   50   60   71   83   96
  21   27   34   42   51   61   72   84   97  111
  28   35   43   52   62   73   85   98  112  127
  36   44   53   63   74   86   99  113  128  144
  45   54   64   75   87  100  114  129  145  162
  55   65   76   88  101  115  130  146  163  181

Suggested by correspondence with Michael Somos.
R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.

Hieronymus Fischer, Table of n, a(n) for n = 1..11401
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences that are permutations of the natural numbers

A self-inverse permutation of the natural numbers.
Cf. A000027, A020703.
Cf. A132666, A132664, A132665, A132674.
Cf. A056011 (boustrophedon).
Cf. A000122, A000700, A010054, A121373.
Cf. A000124, A000217.
Cf. A061579.

a038722 n = a038722_list !! (n-1)
a038722_list = concat a038722_tabl
a038722_tabl = map reverse a000027_tabl
a038722_row n = a038722_tabl !! (n-1)
-- Reinhard Zumkeller, Nov 08 2013

(* Program generates dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := Floor[n+1/2+Sqrt[2n]]
  (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]
(* A038722 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A038722 sequence *)
 (* Clark Kimberling, Jun 06 2011, corrected Jan 26 2025 *)
Table[ n, {m, 12}, {n, m (m + 1)/2, m (m - 1)/2 + 1, -1}] // Flatten (* or *)
Table[ Ceiling[(Sqrt[8 n + 1] - 1)/2]^2 - n + 1, {n, 78}] (* Robert G. Wilson v, Jun 27 2014 *)
With[{nn=20},Reverse/@TakeList[Range[(nn(1+nn))/2],Range[nn]]//Flatten] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Dec 14 2017 *)

a(n)=local(t=floor(1/2+sqrt(2*n))); if(n<1, 0, t^2-n+1) /* Paul D. Hanna */

a(n) = (sqrt(2n-1) - 1/2)*(sqrt(2n-1) + 3/2) - n + 2 = A061579(n-1) + 1. Seen as a square table by antidiagonals, T(n, k) = k + (n+k-1)*(n+k-2)/2, i.e., the transpose of A000027 as a square table.
G.f.: g(x) = (x/(1-x))*(psi(x) - x/(1-x) + 2*Sum_{k>=0} k*x^(k*(k+1)/2)) where psi(x) = Sum_{k>=0} x^(k*(k+1)/2) = (1/2)*x^(-1/8)*theta_2(0,x^(1/2)) is a Ramanujan theta function. - Hieronymus Fischer, Aug 08 2007
a(n) = floor(sqrt(2*n) + 1/2)^2 - n + 1. - Clark Kimberling, Jun 05 2011; corrected by Paul D. Hanna, Jun 27 2011
From Hieronymus Fischer, Apr 30 2012: (Start)
a(n) = a(n-1)-1, if a(n-1)-1 > 0 is not in the set {a(k)| 1<=ka(n) = n for n = 2k(k+1)+1, k >= 0.
a(n+1) = (m+2)(m+3)/2, if 8a(n)-7 is a square of an odd number, otherwise a(n+1) = a(n)-1, where m = (sqrt(8a(n)-7)-1)/2.
a(n) = ceiling((sqrt(8n+1)-1)/2)^2 - n + 1. (End)
G.f. as rectangular array: x*y*(1 - (1 + x)*y + (1 - x + x^2)*y^2)/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Dec 25 2022

A090861 Permutation of natural numbers arising from a spiral.

Original entry on oeis.org

1, 6, 5, 4, 3, 2, 9, 8, 7, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 25, 24, 23, 22, 21, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 49, 48, 47, 46, 45, 44, 43, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50

Offset: 1

Felix Tubiana, Feb 16 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a clockwise spiral: 1 -> 6 -> 5 -> 4 -> 3 -> 2 -> 9 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090915, A090925, A090928, A090929, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[8*x^2-n+2 +x*If[n <= 4*x^2+2*x, -2, 6], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n)=floor((floor(sqrt(n-1)) +1)/2)};
for(n=1,75, print1(8*s(n)^2 -n +2 +s(n)*if(n<= 2*s(n)*(2*s(n)+1), -2, 6), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return 8*x^2 + (-2 if n <= 4*x^2 + 2*x else 6)*x + 2 - n
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A090915 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 8, 7, 6, 5, 4, 3, 2, 9, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 25, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 49, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a square clockwise spiral: 1 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 2 -> 9 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090925, A090928, A090929, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n==(2*x+1)^2, n, 8*x^2 -n+2], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n == (2*s(n)+1)^2, n, 8*s(n)^2-n+2), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n if n == (2*x+1)^2 else 8*x^2 + 2 - n
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A090925 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 9, 2, 3, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 10, 11, 12, 13, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a square counterclockwise spiral: 1 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 2 -> 3 -> 14 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090915, A090928, A090929, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n+2*x <= (2*x+1)^2, n +2*x, n-6*x], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n+2*s(n) <= (2*s(n)+1)^2, n +2*s(n), n - 6*s(n)), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n + 2*x if n + 2*x <= (2*x+1)^2 else n - 6*x
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A090928 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 6, 7, 8, 9, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 24, 25, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 50, 51, 52, 53, 54, 55, 56

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a counterclockwise spiral: 1 -> 6 -> 7 -> 8 -> 9 -> 2 -> 3 -> 4 -> 5 -> 18 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers

Cf. A020703, A090861, A090915, A090925, A090929, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n +4*x <= (2*x+1)^2, n+4*x, n-4*x], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n+4*s(n) <= (2*s(n)+1)^2, n +4*s(n), n - 4*s(n)), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n + 4*x if n + 4*x <= (2*x+1)^2 else n - 4*x
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016

a(n) = A090925(A090925(n)). - Rémy Sigrist, Jul 25 2025

Offset corrected by Eric M. Schmidt, May 18 2016

A090929 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 8, 9, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 74, 75, 76, 77, 78, 79, 80, 81, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a counterclockwise spiral: 1 -> 8 -> 9 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 22 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090915, A090925, A090928, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]] +1)/2]}, Table[If[n +6*x <= (2*x+1)^2, n +6*x, n -2*x], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n+6*s(n) <= (2*s(n)+1)^2, n +6*s(n), n - 2*s(n)), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n + 6*x if n + 6*x <= (2*x+1)^2 else n - 2*x
[a(n) for n in (1..75)]
# Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A090930 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 2, 9, 8, 7, 6, 5, 4, 3, 12, 11, 10, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 30, 29, 28, 27, 26, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 56, 55, 54, 53, 52, 51, 50, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a clockwise spiral: 1 -> 2 -> 9 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 12 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090915, A090925, A090928, A090929.

With[{x = Floor[(Floor[Sqrt[n-1]] +1)/2]}, Table[8*x^2 -n +2 +x*If[n <= 4*x^2 -2*x, -6, 2], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(8*s(n)^2 -n +2 +s(n)*if(n <= 2*s(n)*(2*s(n)-1), -6, 2), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return 8*x^2 + (-6 if n <= 4*x^2 - 2*x else 2)*x + 2 - n
[a(n) for n in (1..75)]
# Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A182194 a(1)=2, a(n)=a(n-1)^2 if the minimal natural number > 1 not yet in the sequence is greater than a(n-1), else a(n)=a(n-1)-1.

Original entry on oeis.org

2, 4, 3, 9, 8, 7, 6, 5, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54

Offset: 1

Hieronymus Fischer, Apr 30 2012

nonn

A reordering of the natural numbers > 1.

The sequence is quasi self-inverse in that a(a(n-1)-1)=n.

			a(2)=4=a(1)^2, since 3>2=a(1) is the minimal number not yet in the sequence (because of a(1)=2);
a(15)=19=a(14)-1, since the minimal number not yet in the sequence (=10) is <=a(14)=20.
a(10^4)=b(8)+b(7)-10^4-2=877.
a(10^6)=b(10)+b(9)-10^6-2= 103539133.

Hieronymus Fischer, Table of n, a(n) for n = 1..10200

Cf. A020703, A038722, A132666, A132664, A132665, A132674, A210882.

a(n)=a(n-1)-1, if a(n-1)-1 > 1 is not in the set {a(k)| 1<=k<=n-1}, else a(n)=a(n-1)^2.
a(a(n)-1)=n+1.
If we define b(1)=2, b(2)=3, b(k)=b(k-2)^2+1, we get the sequence 2, 3, 5, 10, 26, 101, 677, 10202, 458330, 104080805, …. The b(k) are those terms a(n) of the original sequence for which a(n+1)=a(n)^2.
With these b(k) we obtain for k>1:
a(b(k)-2)=b(k-1),
a(b(k)-1)=b(k-1)^2.
a(b(k))=b(k-1)^2 - 1.
a(n)=b(m)+b(m-1)-n-2, where m is the least index such that b(m)>n+1 (valid for n>=1).

A210882 a(1)=1, a(n)=a(n-1)-1 if a(n-1)-1 > 0 is not in the set {a(k)| 1<=k

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 10, 9, 8, 13, 12, 17, 16, 15, 14, 19, 18, 23, 22, 21, 20, 29, 28, 27, 26, 25, 24, 31, 30, 37, 36, 35, 34, 33, 32, 41, 40, 39, 38, 43, 42, 47, 46, 45, 44, 53, 52, 51, 50, 49, 48, 59, 58, 57, 56, 55, 54, 61, 60, 67, 66, 65, 64, 63, 62, 71

Offset: 1

Hieronymus Fischer, Apr 30 2012

nonn

A reordering of the natural numbers.
The sequence is self-inverse in that a(a(n))=n.
If n is a prime, then a(n+1) is the next prime > n. Hence, the subsequence 2, a(2+1), a(a(2+1)+1), a(a(a(2+1)+1)+1), a(a(a(a(2+1)+1)+1)+1), ... generates the sequence of primes A000040.

			a(4)=5, since 5 is the least prime > a(1), a(2), a(3), and the minimal number not yet in the sequence (=4) is greater than 3=a(3).
a(5)=4, since 4 is not in the set {1,2,3,5}={a(k)| 1<=k<n}.
7=p(4)=a(p(3)+1)=a(a(p(2)+1)+1)= a(a(a(p(1)+1)+1)+1)= a(a(a(2+1)+1)+1).

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A020703, A000040, A038722, A132666, A132664, A132665, A132674, A182194.

a(1)=1, a(n)=p (where p is the least prime number > a(k) for 1<=k

a(n)<>n for all n>3.

p(n+1)=a(p(n)+1), where p(n) is the n-th prime.

a(n+1)=p(m+2), if a(n)-1 is the m-th prime, else a(n+1)=a(n)-1, for n>2.

a(n)=p(m)+p(m-1)-n+1, where m is the least index such that p(m)>n-1 (valid for n>2).

Showing 1-9 of 9 results.

cp's OEIS Frontend

A038722 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,2,3,4,... .

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A090861 Permutation of natural numbers arising from a spiral.

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A090915 Permutation of natural numbers arising from a square spiral.

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A090925 Permutation of natural numbers arising from a square spiral.

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A090928 Permutation of natural numbers arising from a square spiral.

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A090929 Permutation of natural numbers arising from a square spiral.

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Mathematica

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Sage

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A090930 Permutation of natural numbers arising from a square spiral.

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