A000027 -id:A000027 - cp's OEIS frontend

A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

1, 4, 14, 47, 156, 517, 1714, 5684, 18851, 62520, 207349, 687676, 2280686, 7563923, 25085844, 83197513, 275925586, 915110636, 3034975799, 10065534960, 33382471801, 110713382644, 367182309614, 1217764693607, 4038731742156, 13394504020957, 44423039068114

Offset: 0

Clark Kimberling, Aug 10 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).
Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).
Guide to p-INVERT sequences using p(S) = 1 - S - S^2:
t(A000012) = t(1,1,1,1,1,1,1,...)    = A001906
t(A000290) = t(1,4,9,16,25,36,...)   = A289779
t(A000027) = t(1,2,3,4,5,6,7,8,...)  = A289780
t(A000045) = t(1,2,3,5,8,13,21,...)  = A289781
t(A000032) = t(2,1,3,4,7,11,14,...)  = A289782
t(A000244) = t(1,3,9,27,81,243,...)  = A289783
t(A000302) = t(1,4,16,64,256,...)    = A289784
t(A000351) = t(1,5,25,125,625,...)   = A289785
t(A005408) = t(1,3,5,7,9,11,13,...)  = A289786
t(A005843) = t(2,4,6,8,10,12,14,...) = A289787
t(A016777) = t(1,4,7,10,13,16,...)   = A289789
t(A016789) = t(2,5,8,11,14,17,...)   = A289790
t(A008585) = t(3,6,9,12,15,18,...)   = A289795
t(A000217) = t(1,3,6,10,15,21,...)   = A289797
t(A000225) = t(1,3,7,15,31,63,...)   = A289798
t(A000578) = t(1,8,27,64,625,...)    = A289799
t(A000984) = t(1,2,6,20,70,252,...)  = A289800
t(A000292) = t(1,4,10,20,35,56,...)  = A289801
t(A002620) = t(1,2,4,6,9,12,16,...)  = A289802
t(A001906) = t(1,3,8,21,55,144,...)  = A289803
t(A001519) = t(1,1,2,5,13,34,...)    = A289804
t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805
t(A008619) = t(1,1,2,2,3,3,4,4,...)  = A289806
t(A080513) = t(1,2,2,3,3,4,4,5,...)  = A289807
t(A133622) = t(1,2,1,3,1,4,1,5,...)  = A289809
t(A000108) = t(1,1,2,5,14,42,...)    = A081696
t(A081696) = t(1,1,3,9,29,97,...)    = A289810
t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843
t(A175676) = t(1,0,0,2,0,0,3,0,...)  = A289844
t(A079977) = t(1,0,1,0,2,0,3,...)    = A289845
t(A059841) = t(1,0,1,0,1,0,1,...)    = A289846
t(A000040) = t(2,3,5,7,11,13,...)    = A289847
t(A008578) = t(1,2,3,5,7,11,13,...)  = A289828
t(A000142) = t(1!, 2!, 3!, 4!, ...)  = A289924
t(A000201) = t(1,3,4,6,8,9,11,...)   = A289925
t(A001950) = t(2,5,7,10,13,15,...)   = A289926
t(A014217) = t(1,2,4,6,11,17,29,...) = A289927
t(A000045*) = t(0,1,1,2,3,5,...)     = A289975 (* indicates prepended 0's)
t(A000045*) = t(0,0,1,1,2,3,5,...)   = A289976
t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977
t(A290990*) = t(0,1,2,3,4,5,...)     = A290990
t(A290990*) = t(0,0,1,2,3,4,5,...)   = A290991
t(A290990*) = t(0,0,01,2,3,4,5,...)  = A290992

			Example 1:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S.
S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - (x + 2x^2 + 3x^3 + 4x^4 + ... )
- p(0) + 1/p(S(x)) = -1 + 1 + x + 3x^2 + 8x^3 + 21x^4 + ...
T(x) = 1 + 3x + 8x^2 + 21x^3 + ...
t(s) = (1,3,8,21,...) = A001906.
***
Example 2:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S - S^2.
S(x) =  x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - ( x + 2x^2 + 3x^3 + 4x^4 + ...) - ( x + 2x^2 + 3x^3 + 4x^4 + ...)^2
- p(0) + 1/p(S(x)) = -1 + 1 + x + 4x^2 + 14x^3 + 47x^4 + ...
T(x) = 1 + 4x + 14x^2 + 47x^3 + ...
t(s) = (1,4,14,47,...) = A289780.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -7, 5, -1)

Cf. A000027.

P:=[1,4,14,47];; for n in [5..10^2] do P[n]:=5*P[n-1]-7*P[n-2]+5*P[n-3]-P[n-4]; od; P; # Muniru A Asiru, Sep 03 2017

z = 60; s = x/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289780 *)

x='x+O('x^99); Vec((1-x+x^2)/(1-5*x+7*x^2-5*x^3+x^4)) \\ Altug Alkan, Aug 13 2017

G.f.: (1 - x + x^2)/(1 - 5 x + 7 x^2 - 5 x^3 + x^4).

a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).

A144112 Weight array W={w(i,j)} of the natural number array A000027.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 11 2008

The lattice lines in the first quadrant (including the x and y axes) cut the plane into unit squares. Suppose a weight w(i,j) is assigned to the square that has as upper right corner the point (i,j). Let s(m,n) be the sum of the weights w(i,j) for 1<=i<=m, 1<=j<=n. We call the array W={w(i,j)} the weight array of the array S={s(m,n)} and S the accumulation array of W. For the case at hand, S is the array of natural numbers having the following antidiagonals: (1), then (2,3), then (4,5,6), then (7,8,9,10) and so on.
Contribution from Clark Kimberling, Sep 14 2008: (Start)
In general, the weight array W of an arbitrary rectangular array S={s(i,j): i>=1, j>=1} is defined in two steps:
(1) extend s by defining s(i,j)=0 if i=0 or j=0;
(2) then w(m,n)=s(m,n)+s(m-1,n-1)-s(m,n-1)-s(m-1,n) for m>=1, n>=1. (End)

			From _Clark Kimberling_, Jan 31 2011: (Start)
Northwest corner:
  1 1 2 3 4 5
  2 1 1 1 1 1
  3 1 1 1 1 1
  4 1 1 1 1 1
  5 1 1 1 1 1.
so that the accumulation array has corner:
  1...2...4...7...11...16
  3...5...8...12..17...23
  6...9...13..18..24...31
  10..14..19..25..32...40
  15..20..26..33..41...50.
s(2,4)=1+1+2+3+2+1+1+1=12. (End)

Stefano Spezia, Table of n, a(n) for n = 1..11325, (first 150 antidiagonals, flattened).

Cf. A000012, A000027.

Array[Append[PadRight[{#},#,1],#+1]&,15,0] (* Paolo Xausa, Dec 21 2023 *)

Row 1: 1 followed by A000027.
Row n: n followed by A000012, for n>1.
G.f.: x*y*(1 - (1 + x)*y + (1 - x + x^2)*y^2)/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Oct 01 2023

A336811 Irregular triangle read by rows T(n,k) in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive integers A000027, with n >= 1 and k >= 1.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 1, 5, 3, 2, 1, 1, 6, 4, 3, 2, 2, 1, 1, 7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 10, 8, 7, 6, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 20 2020

In other words: row n lists A028310(n-1) blocks where the m-th block consists of A187219(m) copies of n - m + [m=1], with n >= 1 and m >= 1, where [] is the Iverson bracket. [Corrected by Paolo Xausa, Feb 10 2023]
All divisors of all terms in row n are also all parts in the last section of the set of partitions of n.
Thus all divisors of all terms of the first n rows of triangle are also all parts of all partitions of n. In other words: all divisors of the first A000070(n-1) terms of the sequence are also all parts of all partitions of n. - Omar E. Pol, Jun 19 2021
From Omar E. Pol, Jul 31 2021: (Start)
The number of k's in row n is equal to A002865(n-k), 1 <= k <= n.
The number of terms >= k in row n is equal to A000041(n-k), 1 <= k <= n.
The number of k's in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000041(n-k), 1 <= k <= n.
The number of terms >= k in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n.
First n rows of triangle (or first A000070(n-1) terms of the sequence) in nonincreasing order give the n-th row of A176206. (End)

			Triangle begins:
1;
2;
3, 1;
4, 2, 1;
5, 3, 2, 1, 1;
6, 4, 3, 2, 2, 1, 1;
7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1;
8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1;
9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
...
For n = 6, by definition the length of row 6 is A000041(6-1) = A000041(5) = 7, so the row 6 of triangle has seven terms. Since every column lists the positive integers A000027 so the row 6 is [6, 4, 3, 2, 2, 1, 1].
Then we have that the divisors of the numbers of the 6th row are:
.
6th row of the triangle ---------->   6 4 3 2 2 1 1
                                      3 2 1 1 1
                                      2 1
                                      1
.
There are seven 1's, four 2's, two 3's, one 4 and one 6.
In total there are 7 + 4 + 2 + 1 + 1 = 15 divisors.
On the other hand the last section of the set of the partitions of 6 can be represented in several ways, five of them as shown below:
._ _ _ _ _ _
|_ _ _      |       6    6                  6                       6
|_ _ _|_    |     3 3    3 3              3   3                     3   3
|_ _    |   |     4 2    4 2            4       2                     4     2
|_ _|_ _|_  |   2 2 2    2 2 2        2   2       2                 2 2   2
          | |       1      1                        1                           1
          | |       1        1                        1                       1
          | |       1        1                          1                   1
          | |       1          1                          1               1
          | |       1          1                            1           1
          | |       1            1                            1       1
          |_|       1              1                            1   1
.
   Figure 1.  Figure 2.  Figure 3.        Figure 4.                   Figure 5.
.
In every figure there are seven 1's, four 2's, two 3's, one 4 and one 6, as shown also the 6th row of A182703.
In total there are 7 + 4 + 2 + 1 + 1 = A138137(6) = 15 parts in every figure.
Figure 5 is an arrangement that shows the correspondence between divisors and parts since the columns give the divisors of the terms of 6th row of triangle.
Finally we can see that all divisors of all numbers in the 6th row of the triangle are the same positive integers as all parts in the last section of the set of the partitions of 6.
Example edited by _Omar E. Pol_, Aug 10 2021

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened).

Row sums give A000070.
Row n has length A000041(n-1).
Every column k gives A000027.
Companion of A176206.
Cf. A000007, A000041, A027750, A028310, A002865, A133735, A135010, A138121, A138137, A182703, A187219, A207378, A221529, A336812, A339278, A340035, A340061, A346741.

A336811[row_]:=Flatten[Table[ConstantArray[row-m,PartitionsP[m]-PartitionsP[m-1]],{m,0,row-1}]];
Array[A336811,10] (* Generates 10 rows *) (* Paolo Xausa, Feb 10 2023 *)

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--;); 1+s;}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

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

Original entry on oeis.org

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

A136119 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + a(n).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 95, 96, 97, 99, 100

Offset: 1

Ctibor O. Zizka, Mar 16 2008

Apparently a(n) = A001953(n-1)+1 = floor((n-1/2)*sqrt(2))+1 (confirmed for n < 20000) and a(n+1) - a(n) = A001030(n). From the definitions these conjectures are by no means obvious. Can they be proved? - Klaus Brockhaus, Apr 15 2008 [For an affirmative answer, see the Cloitre link.]

This is the s(n)-Wythoff sequence for s(n)=2n-1; see A184117 for the definition. Complement of A184119. - Clark Kimberling, Jan 09 2011

			First few steps are:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 1; delete term at position 1+a(1) = 2: 2;
1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 2; delete term at position 2+a(2) = 5: 6;
1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 3; delete term at position 3+a(3) = 7: 9;
1,3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,20,...
n = 4; delete term at position 4+a(4) = 9: 12;
1,3,4,5,7,8,10,11,13,14,15,16,17,18,19,20,...
n = 5; delete term at position 5+a(5) = 12: 16;
1,3,4,5,7,8,10,11,13,14,15,17,18,19,20,...
n = 6; delete term at position 6+a(6) = 14: 19;
1,3,4,5,7,8,10,11,13,14,15,17,18,20,...

B. Cloitre, The golden sieve, preprint 2008

Alois P. Heinz, Table of n, a(n) for n = 1..1000
D. X. Charles, Sieve Methods, July 2000, University of Wisconsin.
Benoit Cloitre, On the proof of Klaus Brockhaus's conjectures
R. Eismann, Decomposition of natural numbers into weight X level + jump and application to a new classification of prime numbers, arXiv:0711.0865 [math.NT], 2007-2010.
M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp.41-56.
Index entries for sequences generated by sieves

Cf. A000027, A001953 (floor((n+1/2)*sqrt(2))), A001030 (fixed under 1 -> 21, 2 -> 211), A136110, A137292.
Cf. A000959, A099267.
Cf. A242535.
Cf. A000217 (T).

import Data.List (delete)
a136119 n = a136119_list !! (n-1)
a136119_list = f [1..] where
   f zs@(y:xs) = y : f (delete (zs !! y) xs)
-- Reinhard Zumkeller, May 17 2014

[Ceiling((n-1/2)*Sqrt(2)): n in [1..100]]; // Vincenzo Librandi, Jul 01 2019

f[0] = Range[100]; f[n_] := f[n] = Module[{pos = n + f[n-1][[n]]}, If[pos > Length[f[n-1]], f[n-1], Delete[f[n-1], pos]]]; f[1]; f[n = 2]; While[f[n] != f[n-1], n++]; f[n] (* Jean-François Alcover, May 08 2019 *)
T[n_] := n (n + 1)/2; Table[1 + 2 Sqrt[T[n-1]] , {n, 1, 71}] // Floor (* Ralf Steiner, Oct 23 2019 *)

apply( {A136119(n)=sqrtint(n*(n-1)*2)+1}, [1..99]) \\ M. F. Hasler, Jul 04 2022

a(n) = ceiling((n-1/2)*sqrt(2)). This can be proved in the same way as the formula given for A099267. There are some generalizations. For instance, it is possible to consider "a(n)+K*n" instead of "a(n)+n" for deleting terms where K=0,1,2,... is fixed. The constant involved in the Beatty sequence for the sequence of deleted terms then depends on K and equals (K + 1 + sqrt((K+1)^2 + 4))/2. K=0 is related to A099267. 1+A001954 is the complement sequence of this sequence A136119. - Benoit Cloitre, Apr 18 2008
a(n) = floor(1 + 2*sqrt(T(n-1))), with triangular numbers T(). - Ralf Steiner, Oct 23 2019
Lim_{n->inf}(a(n)/(n - 1)) = sqrt(2), with {a(n)/(n - 1)} decreasing. - Ralf Steiner, Oct 24 2019

Edited and extended by Klaus Brockhaus, Apr 15 2008

An incorrect g.f. removed by Alois P. Heinz, Dec 14 2012

A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 2, 18, 2, 23, 2, 59, 7, 23, 2, 94, 2, 23, 16, 195, 2, 80, 2, 94, 16, 23, 2, 355, 7, 23, 29, 94, 2, 467, 2, 672, 16, 23, 16, 706, 2, 23, 16, 355, 2, 467, 2, 94, 67, 23, 2, 1331, 7, 80, 16, 94, 2, 302, 16, 355, 16, 23, 2, 1894, 2, 23, 67, 2422, 16, 467, 2, 94, 16, 467, 2, 2779, 2, 23, 67, 94, 16, 467, 2, 1331, 121, 23, 2, 1894, 16, 23, 16, 355, 2, 1832

Offset: 1

Antti Karttunen, May 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000027, A001511, A046523, A286160, A286162, A286163, A286164, A286251.

A001511(n) = (1+valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286161(n) = (2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286161.txt", n, " ", A286161(n)));

from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a046523(n)) # Indranil Ghosh, May 06 2017

(define (A286161 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A046523 n)) 2) (- (A001511 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n)).

A291750 Compound filter: a(n) = P(A003557(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 4, 7, 8, 16, 67, 29, 19, 18, 154, 67, 80, 92, 277, 277, 53, 154, 94, 191, 173, 497, 631, 277, 109, 50, 862, 75, 302, 436, 2557, 497, 169, 1129, 1432, 1129, 142, 704, 1771, 1541, 214, 862, 4561, 947, 668, 328, 2557, 1129, 179, 98, 236, 2557, 905, 1432, 199, 2557, 355, 3161, 4006, 1771, 2630, 1892, 4561, 564, 593, 3487, 10297, 2279, 1487, 4561, 10297, 2557

Offset: 1

Antti Karttunen, Sep 04 2017

nonn

A000203 (sigma(n)) is a function of this sequence, because formula
  A000203(n) = A092261(n) * A295294(n)
can be rewritten as a relation:
  A000203(n) = A000203(A057521(n)) * A048250(n) / A048250(A057521(n)),
where A057521(n) = A064549(A003557(n)), thus A000203(n) is a function of A003557(n) and A048250(n), the values that are packed here into a(n).
A001615 (Dedekind's psi) is a function of this sequence, because it can be written as A001615(n) = A003557(n)*A048250(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000027, A000203, A001615, A003557, A048250, A291751 (rgs-version of this filter).

A003557(n) = n/factorback(factor(n)[, 1]); \\ This function from Charles R Greathouse IV, Nov 17 2014
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));

a(n) = (1/2)*(2 + ((A003557(n) + A048250(n))^2) - A003557(n) - 3*A048250(n)).

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

Original entry on oeis.org

1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375

Offset: 1

Clark Kimberling, Feb 03 2011

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
  1....2.....4.....7...11...16...22...29...
  3....5.....8....12...17...23...30...38...
  6....9....13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787.  Deleting the main diagonal and then summing give A185787.  Analogous treatments to the left of the main diagonal give A100182 and A101165.  Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1:  A000124
row 2:  A022856
row 3:  A016028
row 4:  A145018
row 5:  A077169
col 1:  A000217
col 2:  A000096
col 3:  A034856
col 4:  A055998
col 5:  A046691
col 6:  A052905
col 7:  A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k.  Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000027, A185788, A100182, A101165, A079824.

[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}]  (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

a(n)=n*(7*n^2-6*n+5)/6.

G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Edited by Clark Kimberling, Feb 25 2023

A286160 Compound filter: a(n) = T(A000010(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 12, 14, 23, 27, 59, 42, 40, 65, 109, 90, 61, 86, 261, 152, 142, 189, 179, 148, 115, 275, 473, 273, 148, 318, 265, 434, 674, 495, 1097, 320, 226, 430, 1093, 702, 271, 430, 757, 860, 832, 945, 485, 619, 373, 1127, 1969, 1032, 485, 698, 619, 1430, 838, 1030, 1105, 856, 556, 1769, 2791, 1890, 625, 1117, 4497, 1426, 1196, 2277, 935, 1220

Offset: 1

Antti Karttunen, May 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000010, A000027, A046523, A286161, A286162, A286163, A286164.

Cf. for example A061468 (one of the sequences this matches with).

A000010(n) = eulerphi(n);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286160(n) = (2 + ((A000010(n)+A046523(n))^2) - A000010(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286160.txt", n, " ", A286160(n)));

from sympy import factorint, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(totient(n), a046523(n)) # Indranil Ghosh, May 06 2017

(define (A286160 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A046523 n)) 2) (- (A000010 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A000010(n)+A046523(n))^2) - A000010(n) - 3*A046523(n)).

A286162 Compound filter: a(n) = T(A001511(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 7, 9, 16, 12, 29, 14, 16, 23, 67, 18, 67, 38, 121, 20, 16, 23, 67, 31, 436, 80, 277, 25, 67, 80, 631, 48, 277, 138, 497, 27, 16, 23, 67, 31, 436, 80, 277, 40, 436, 467, 1771, 94, 1771, 302, 1129, 33, 67, 80, 631, 94, 1771, 668, 2557, 59, 277, 302, 2557, 156, 1129, 530, 2017, 35, 16, 23, 67, 31, 436, 80, 277, 40, 436, 467, 1771, 94, 1771, 302, 1129, 50

Offset: 1

Antti Karttunen, May 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A001511, A278222, A286160, A286161, A286163, A286164.

A001511(n) = (1+valuation(n,2));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A286162(n) = (2 + ((A001511(n)+A278222(n))^2) - A001511(n) - 3*A278222(n))/2;
for(n=1, 10000, write("b286162.txt", n, " ", A286162(n)));

from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a278222(n): return a046523(a005940(n + 1))
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def a(n): return T(a001511(n), a278222(n)) # Indranil Ghosh, May 05 2017

(define (A286162 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A278222 n)) 2) (- (A001511 n)) (- (* 3 (A278222 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A278222(n))^2) - A001511(n) - 3*A278222(n)).

Showing 1-10 of 2207 results. Next

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A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

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A144112 Weight array W={w(i,j)} of the natural number array A000027.

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A336811 Irregular triangle read by rows T(n,k) in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive integers A000027, with n >= 1 and k >= 1.

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

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A136119 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + a(n).

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A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A291750 Compound filter: a(n) = P(A003557(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

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