A003602 -id:A003602 - cp's OEIS frontend

A091067 Numbers whose odd part is of the form 4k+3.

3, 6, 7, 11, 12, 14, 15, 19, 22, 23, 24, 27, 28, 30, 31, 35, 38, 39, 43, 44, 46, 47, 48, 51, 54, 55, 56, 59, 60, 62, 63, 67, 70, 71, 75, 76, 78, 79, 83, 86, 87, 88, 91, 92, 94, 95, 96, 99, 102, 103, 107, 108, 110, 111, 112, 115, 118, 119, 120, 123, 124, 126, 127, 131

Offset: 1

Ralf Stephan, Feb 22 2004

Either of form 2*a(m) or 4k+3, k >= 0, 0 < m < n.
A000265(a(n)) is an element of A004767.
a(n) such that A038189(a(n)) = 1.
Numbers n such that Kronecker(-n, m) = Kronecker(m, n) for all m. - Michael Somos, Sep 22 2005
From Antti Karttunen, Feb 20-21 2015: (Start)
Gives all n for which A005811(n) - A005811(n-1) = -1, from which follows that a(n) = the least k such that A255070(k) = n.
Gives the positions of even terms in A003602. (End)
Indices of negative terms in A164677. - M. F. Hasler, Aug 06 2015
Indices of the 0's in A014577. - Gabriele Fici, Jun 02 2016
Also indices of -1 in A034947. - Jianing Song, Apr 24 2021
Conjecture: alternate definition of same sequence is that a(1)=3 and a(n) is the smallest number > a(n-1) so that no number that is the sum of at most 2 terms in this sequence is a power of 2. - J. Lowell, Jan 20 2024
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Aug 31 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, On three conjectures of P. Barry, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Kevin Ryde, Iterations of the Dragon Curve, see index TurnRight, with a(n) = TurnRight(n-1).

Essentially one less than A060833.
Characteristic function: A038189.
Complement of A091072.
First differences are in A106836 (from its second term onward).
Sequence A246590 gives the even terms.
Gives the positions of records (after zero) for A255070 (equally, the position of the first n there).
Cf. A106837 (gives n such that both n and n+1 are terms of this sequence).
Cf. A098502 (gives n such that both n and n+2 are, but n+1 is not in this sequence).
Cf. also A000265, A003602, A004767, A005811, A014707, A034947, A055975, A236840, A255068, A255327, A255330.

import Data.List (elemIndices)
a091067 n = a091067_list !! (n-1)
a091067_list = map (+ 1) $ elemIndices 1 a014707_list
-- Reinhard Zumkeller, Sep 28 2011
(Scheme, with Antti Karttunen's IntSeq-library, two versions)
(define A091067 (MATCHING-POS 1 1 (COMPOSE even? A003602)))
(define A091067 (NONZERO-POS 1 0 A038189))
;; Antti Karttunen, Feb 20 2015

Select[Range[150], Mod[# / 2^IntegerExponent[#, 2], 4] == 3 &] (* Amiram Eldar, Aug 31 2024 *)

for(n=1,200,if(((n/2^valuation(n,2)-1)/2)%2,print1(n",")))

{a(n) = local(m, c); if( n<1, 0, c=0; m=1; while( cMichael Somos, Sep 22 2005 */

is_A091067(n)=bittest(n,valuation(n,2)+1) \\ M. F. Hasler, Aug 06 2015

a(n) = my(t=1); n<<=1; forstep(i=logint(n,2),0,-1, if(bittest(n,i)==t, n++;t=!t)); n; \\ Kevin Ryde, Mar 21 2021

a(n) = A060833(n+1) - 1. [See N. Sato's Feb 12 2013 comment in A060833.]
Other identities. For all n >= 1 it holds that:
A014707(a(n) + 1) = 1. - Reinhard Zumkeller, Sep 28 2011
A055975(a(n)) < 0. - Reinhard Zumkeller, Apr 28 2012
From Antti Karttunen, Feb 20-21 2015: (Start)
a(n) = A246590(n)/2.
A255070(a(n)) = n, or equally, A236840(a(n)) = 2n.
a(n) = 1 + A255068(n-1). (End)

A078719 Number of odd terms among n, f(n), f(f(n)), ...., 1 for the Collatz function (that is, until reaching "1" for the first time), or -1 if 1 is never reached.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 6, 1, 7, 2, 5, 3, 3, 6, 6, 1, 4, 7, 7, 2, 2, 5, 5, 3, 8, 3, 42, 6, 6, 6, 40, 1, 9, 4, 4, 7, 7, 7, 12, 2, 41, 2, 10, 5, 5, 5, 39, 3, 8, 8, 8, 3, 3, 42, 42, 6, 11, 6, 11, 6, 6, 40, 40, 1, 9, 9, 9, 4, 4, 4, 38, 7, 43, 7, 4, 7, 7, 12, 12, 2, 7, 41, 41, 2, 2, 10, 10, 5, 10, 5, 34, 5, 5, 39

Offset: 1

Joseph L. Pe, Dec 20 2002

nonn

The Collatz function (related to the "3x+1 problem") is defined by: f(n) = n/2 if n is even; f(n) = 3n + 1 if n is odd. A famous conjecture states that n, f(n), f(f(n)), .... eventually reaches 1.
a(n) = A006667(n) + 1; a(A000079(n))=1; a(A062052(n))=2; a(A062053(n))=3; a(A062054(n))=4; a(A062055(n))=5; a(A062056(n))=6; a(A062057(n))=7; a(A062058(n))=8; a(A062059(n))=9; a(A062060(n))=10. - Reinhard Zumkeller, Oct 08 2011
The count includes also the starting value n if it is odd. See A286380 for the version which never includes n itself. - Antti Karttunen, Aug 10 2017

			The terms n, f(n), f(f(n)), ...., 1 for n = 12 are: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, of which 3 are odd. Hence a(12) = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Chris K. Caldwell and G. L. Honaker, Jr., Prime curio for 41 (which says 41 is a fixed point)
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A014682, A078720, A139391, A286380.

Cf. A258145, A003602.

a078719 =
   (+ 1) . length . filter odd . takeWhile (> 2) . (iterate a006370)
a078719_list = map a078719 [1..]
-- Reinhard Zumkeller, Oct 08 2011

a:= proc(n) option remember; `if`(n<2, 1,
      `if`(n::even, a(n/2), 1+a(3*n+1)))
    end:
seq(a(n), n=1..94);  # Alois P. Heinz, Jan 17 2025

f[n_] := Module[{a, i, o}, i = n; o = 1; a = {}; While[i > 1, If[Mod[i, 2] == 1, o = o + 1]; a = Append[a, i]; i = f[i]]; o]; Table[f[i], {i, 1, 100}]
Table[Count[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &], ?OddQ], {n, 94}] (* _Jayanta Basu, Jun 15 2013 *)

a(n) = {my(x=n, v=List([])); while(x>1, if(x%2==0, x=x/2, listput(v, x); x=3*x+1)); 1+#v;} \\ Jinyuan Wang, Dec 29 2019

a(n) = A286380(n) + A000035(n). - Antti Karttunen, Aug 10 2017

a(n) = A258145(A003602(n)-1). - Alan Michael Gómez Calderón, Sep 15 2024

"Escape clause" added to definition by N. J. A. Sloane, Jun 06 2017

A035528 Euler transform of A027656(n-1).

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 9, 13, 19, 28, 42, 57, 84, 115, 164, 227, 313, 429, 588, 799, 1079, 1461, 1952, 2617, 3480, 4627, 6111, 8072, 10604, 13905, 18181, 23701, 30828, 39990, 51763, 66822, 86124, 110687, 142039, 181841, 232409, 296401, 377419, 479635, 608558, 770818

Offset: 0

Christian G. Bower

nonn

Also the weigh transform of A003602. - John Keith, Nov 17 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000219, A003602, A052847, A263150, A263352, A262876, A263136, A263141.

nmax = 50; CoefficientList[Series[-1 + Product[1/(1 - x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 100; Flatten[{0, Rest[CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Oct 10 2015 *)

a(n) ~ A^(1/2) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) + Pi^2 * n^(1/3)/(3*2^(8/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(4/3)) / (sqrt(3*Pi) * 2^(71/72) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 02 2015

A135764 Distribute the natural numbers in columns based on the occurrence of "2" in each prime factorization; square array A(row,col) = 2^(row-1) * ((2*col)-1), read by descending antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 11, 18, 28, 40, 48, 32, 13, 22, 36, 56, 80, 96, 64, 15, 26, 44, 72, 112, 160, 192, 128, 17, 30, 52, 88, 144, 224, 320, 384, 256, 19, 34, 60, 104, 176, 288, 448, 640, 768, 512, 21, 38, 68, 120, 208, 352, 576, 896, 1280, 1536, 1024, 23, 42, 76, 136, 240, 416, 704, 1152, 1792, 2560, 3072, 2048, 25, 46, 84, 152, 272, 480, 832, 1408, 2304, 3584, 5120, 6144, 4096, 27, 50, 92, 168, 304, 544, 960, 1664, 2816

Offset: 1

Alford Arnold, Nov 29 2007

The array in A135764 is identical to the array in A054582 [up to the transposition and different indexing. - Clark Kimberling, Dec 03 2010; comment amended by Antti Karttunen, Feb 03 2015; please see the illustration in Example section].
The array gives a bijection between the natural numbers N and N^2. A more usual bijection is to take the natural numbers A000027 and write them in the usual OEIS square array format. However this bijection has the advantage that it can be formed by iterating the usual bijection between N and 2N. - Joshua Zucker, Nov 04 2011
The array can be used to determine the configurations of k-th Towers of Hanoi moves, by labeling odd row terms C,B,A,C,B,A,... and even row terms B,C,A,B,C,A,.... Then given k equal to or greater than term "a" in each n-th row, but less than the next row term, record the label A, B, or C for term "a". This denotes the peg position for the disc corresponding to the n-th row. For example, with k = 25, five discs are in motion since the binary for 25 = 11001, five bits. We find that 25 in row 5 is greater than 16 labeled C, but less than 48. Thus, disc 5 is on peg C. In the 4th row, 25 is greater than 24 (a C), but less than 40, so goes onto the C peg. Similarly, disc 3 is on A, 2 is on A, and disc 1 is on A. Thus, discs 2 and 3 are on peg A, while 1, 4, and 5 are on peg C. - Gary W. Adamson, Jun 22 2012
Shares with arrays A253551 and A254053 the property that A001511(n) = k for all terms n on row k and when going downward in each column, terms grow by doubling. - Antti Karttunen, Feb 03 2015
Let P be the infinite palindromic word having initial word 0 and midword sequence (1,2,3,4,...) = A000027. Row n of the array A135764 gives the positions of n-1 in S. ("Infinite palindromic word" is defined at A260390.) - Clark Kimberling, Aug 13 2015
The probability distribution series 1 = 2/3 + 4/15 + 16/255 + 256/65535 + ... + A001146(n-1)/A051179(n) governs the proportions of terms in A001511 from row n of the array. In A001511(1..15) there are ((2/3) * 15) = ten terms from row one of the array, ((4/15) * 15) = four terms from row two, and ((16/255) * 15) = one (rounded), giving one term from row three (a 4). - Gary W. Adamson, Dec 16 2021
From Gary W. Adamson, Dec 30 2021: (Start)
Subarrays representing the number of divisors of an integer can be mapped on the table. For 60, write the odd divisors on the top row: 1, 3, 5, 15. Since 60 has 12 divisors, let the left column equal 1, 2, 4, where 4 is the highest power of 2 dividing 60. Multiplying top row terms by left column terms, we get the result:
  1   3   5  15
  2   6  10  30
  4  12  20  60. (End)

			The table begins
   1,  3,   5,   7,   9,  11,  13,  15,  17,  19,  21,  23, ...
   2,  6,  10,  14,  18,  22,  26,  30,  34,  38,  42,  46, ...
   4, 12,  20,  28,  36,  44,  52,  60,  68,  76,  84,  92, ...
   8, 24,  40,  56,  72,  88, 104, 120, 136, 152, 168, 184, ...
  16, 48,  80, 112, 144, 176, 208, 240, 272, 304, 336, 368, ...
  32, 96, 160, 224, 288, 352, 416, 480, 544, 608, 672, 736, ...
etc.
For n = 6, we have [A002260(6), A004736(6)] = [3, 1] (i.e., 6 corresponds to location 3,1 (row,col) in above table) and A(3,1) = A000079(3-1) * A005408(1-1) = 2^2 * 1 = 4.
For n = 13, we have [A002260(13), A004736(13)] = [3, 3] (13 corresponds to location 3,3 (row,col) in above table) and A(3,3) = A000079(3-1) * A005408(3-1) = 2^2 * 5 = 20.
For n = 23, we have [A002260(23), A004736(23)] = [2, 6] (23 corresponds to location 2,6) and A(2,6) = A000079(2-1) * A005408(6-1) = 2^1 * 11 = 22.

Transpose: A054582.
Inverse permutation: A249725.
Column 1: A000079.
Row 1: A005408.
Cf. A001511 (row index), A003602 (column index, both one-based).
Related arrays: A135765, A253551, A254053, A254055.
Cf. A000027, A002260, A004736, A064989, A135766, A249746.
Cf. also permutations A246675, A246676, A249741, A249811, A249812.
Cf. A260390.
Cf. A001146, A051179.

seq(seq(2^(j-1)*(2*(i-j)+1),j=1..i),i=1..20); # Robert Israel, Feb 03 2015

f[n_] := Block[{i, j}, {1}~Join~Flatten@ Last@ Reap@ For[j = 1, j <= n, For[i = j, i > 0, Sow[2^(j - i - 1)*(2 i + 1)], i--], j++]]; f@ 10 (* Michael De Vlieger, Feb 03 2015 *)

a(n) = {s = ceil((1 + sqrt(1 + 8*n)) / 2); r = n - binomial(s-1, 2) - 1;k = s - r - 2; 2^r * (2 * k + 1) } \\ David A. Corneth, Feb 05 2015

(define (A135764 n) (A135764bi (A002260 n) (A004736 n)))
(define (A135764bi row col) (* (A000079 (- row 1)) (+ -1 col col)))
;; Antti Karttunen, Feb 03 2015

From Antti Karttunen, Feb 03 2015: (Start)
A(row, col) = 2^(row-1) * ((2*col)-1) = A000079(row-1) * A005408(col-1).
A(row,col) = A064989(A135765(row,A249746(col))).
A(row,col) = A(row+1,col)/2 [discarding the topmost row and halving the rest of terms gives the array back].
A(row,col) = A(row,col+1) - A000079(row) [discarding the leftmost column and subtracting 2^{row number} from the rest of terms gives the array back].
(End)
G.f.: ((2*x+1)*Sum_{i>=0} 2^i*x^(i*(i+1)/2) + 2*(1-2*x)*Sum_{i>=0} i*x^(i*(i+1)/2) + (1-6*x)*Sum_{i>=0} x^(i*(i+1)/2) - 1 - 2*x)*x/(1-2*x)^2. These sums are related to Jacobi theta functions. - Robert Israel, Feb 03 2015

More terms from Sean A. Irvine, Nov 23 2010

Name amended and the illustration of array in the example section transposed by Antti Karttunen, Feb 03 2015

A103391 "Even" fractal sequence for the natural numbers: Deleting every even-indexed term results in the same sequence.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 40, 21, 41, 4, 42, 22, 43, 12, 44, 23, 45, 7, 46, 24, 47, 13, 48, 25, 49, 3, 50, 26, 51, 14, 52, 27, 53, 8

Offset: 1

Eric Rowland, Mar 20 2005

A003602 is the "odd" fractal sequence for the natural numbers.

Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(A005940(i)) = A348717(A005940(j)) for all i, j >= 1. A365718 is an analogous sequence related to A356867 (Doudna variant D(3)). - Antti Karttunen, Sep 17 2023

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..10000 from Reinhard Zumkeller)

Cf. A003602, A005940, A025480, A220466, A286387, A353368 (Dirichlet inverse).
Cf. also A110962, A110963, A365718.
Differs from A331743(n-1) for the first time at n=192, where a(192) = 97, while A331743(191) = 23.
Differs from A351460.

-- import Data.List (transpose)
a103391 n = a103391_list !! (n-1)
a103391_list = 1 : ks where
   ks = concat $ transpose [[2..], ks]
-- Reinhard Zumkeller, May 23 2013

nmax := 82: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 2 to ceil(nmax/(p+2))+1 do a((2*n-3)*2^p+1) := n od: od: a(1) := 1: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 28 2013

a[n_] := ((n-1)/2^IntegerExponent[n-1, 2] + 3)/2; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

A003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1))); \\ Antti Karttunen, Feb 05 2020

def v(n): b = bin(n); return len(b) - len(b.rstrip("0"))
def b(n): return (n//2**v(n)+1)//2
def a(n): return 1 if n == 1 else 1 + b(n-1)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, May 29 2022

def A103391(n): return (n-1>>(n-1&-n+1).bit_length())+2 if n>1 else 1 # Chai Wah Wu, Jan 04 2024

For n > 1, a(n) = A003602(n-1) + 1. - Benoit Cloitre, May 26 2007, indexing corrected by Antti Karttunen, Feb 05 2020
a((2*n-3)*2^p+1) = n, p >= 0 and n >= 2, with a(1) = 1. - Johannes W. Meijer, Jan 28 2013
Sum_{k=1..n} a(k) ~ n^2/6. - Amiram Eldar, Sep 24 2023

Data section extended up to a(105) (to better differentiate from several nearby sequences) by Antti Karttunen, Feb 05 2020

A113415 Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 5, 1, 8, 4, 7, 3, 8, 5, 14, 1, 10, 8, 11, 4, 18, 7, 13, 3, 17, 8, 22, 5, 16, 14, 17, 1, 26, 10, 26, 8, 20, 11, 30, 4, 22, 18, 23, 7, 42, 13, 25, 3, 30, 17, 38, 8, 28, 22, 38, 5, 42, 16, 31, 14, 32, 17, 55, 1, 44, 26, 35, 10, 50, 26, 37, 8, 38, 20, 65, 11, 50, 30, 41

Offset: 1

Michael Somos, Oct 29 2005

nonn

Arithmetic mean between the number of odd divisors (A001227) and their sum (A000593). This fact was essentially found by the algorithmic search of Jon Maiga's Sequence Machine, and is easily seen to be correct when compared to the PARI-program given by the original author. - Antti Karttunen, Dec 07 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A113415, Sequence Machine.

Cf. A000265, A000593, A001227, A001511, A003602, A048673, A064989, A069734, A336840, A349371, A349915 (Dirichlet inverse).

Quadrisection of A349916.

Array[DivisorSum[#, If[OddQ[#], (# + 1)/2, 0] &] &, 79] (* Michael De Vlieger, Dec 08 2021 *)

a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)))

G.f.: Sum_{k>0} x^k/(1-x^(2k))^2 = Sum_{k>0} k x^(2k-1)/(1-x^(2k-1)).
a(n) = (1/2) * Sum_{d|n} (d+1)*(d mod 2). - Wesley Ivan Hurt, Nov 25 2021 [From PARI prog]
From Antti Karttunen, Dec 07 2021: (Start)
All these formulas, except the last, were found by the Sequence Machine in some form or another:
a(n) = (1/2) * (A000593(n)+A001227(n)).
a(n) = A069734(A000265(n)). [See either Rutherford's or Luschny's formula in A069734]
a(n) = A349371(n) / A001511(n).
a(n) = A349371(A000265(n)) = A336840(A064989(n)).
a(n) = a(2*n) = a(A000265(n)) = A349916(4*n).
(End)

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

Original entry on oeis.org

1, 3, 6, 5, 10, 20, 7, 14, 28, 56, 9, 18, 36, 72, 144, 11, 22, 44, 88, 176, 352, 13, 26, 52, 104, 208, 416, 832, 15, 30, 60, 120, 240, 480, 960, 1920, 17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 21, 42, 84, 168

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Central terms give A118415; row sums give A118414;
T(n,1) = A005408(n-1);
T(n,2) = A016825(n-1) for n>1;
T(n,3) = A017113(n-1) for n>2;
T(n,4) = A051062(n-1) for n>3;
T(n,n-2) = A052951(n-1) for n>2;
T(n,n) = A014480(n-1) = A118416(n,n);
A001511(T(n,k)) = A002260(n,k);
A003602(T(n,k)) = A002024(n,k).
G.f.: x*y*(1 + x + 2*x*y - 6*x^2*y)/((1 - x)^2*(1 - 2*x*y)^2). - Stefano Spezia, Dec 22 2024

			   1
   3   6
   5  10  20
   7  14  28  56
   9  18  36  72 144
  11  22  44  88 176 352
  13  26  52 104 208 416  832
  15  30  60 120 240 480  960 1920
  17  34  68 136 272 544 1088 2176 4352
  19  38  76 152 304 608 1216 2432 4864 9728
  ...

Cf. A001511, A002024, A002260, A003602, A005408, A014480, A016825, A017113, A051062, A052951, A117303, A118414, A118415, A118416, A117303.

Select[Flatten[Table[(2n-1)2^(k-1),{n,20},{k,0,n}]],IntegerQ] (* Harvey P. Dale, Jan 17 2024 *)

from math import isqrt, comb
def A118413(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    return ((a<<1)-1)<Chai Wah Wu, Jun 20 2025

A118416 Triangle read by rows: T(n,k) = (2k-1)2^(n-1), 0 < k <= n.

Original entry on oeis.org

1, 2, 6, 4, 12, 20, 8, 24, 40, 56, 16, 48, 80, 112, 144, 32, 96, 160, 224, 288, 352, 64, 192, 320, 448, 576, 704, 832, 128, 384, 640, 896, 1152, 1408, 1664, 1920, 256, 768, 1280, 1792, 2304, 2816, 3328, 3840, 4352, 512, 1536, 2560, 3584, 4608, 5632, 6656, 7680

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Row sums give A014477: Sum_{k=1..n} T(n,k) = A014477(n-1);
central terms give A118415; T(2*k-1,k) = A058962(k-1);
T(n,1) = A000079(n-1);
T(n,2) = A007283(n-1) for n > 1;
T(n,3) = A020714(n-1) for n > 2;
T(n,4) = A005009(n-1) for n > 3;
T(n,5) = A005010(n-1) for n > 4;
T(n,n-1) = A118417(n-1) for n > 1;
T(n,n) = A014480(n-1) = A118413(n,n);
A001511(T(n,k)) = A002024(n,k);
A003602(T(n,k)) = A002260(n,k).
The alternating row sums, Sum_{k=1..n} (-1)^(k+1)*T(n,k), are: (a) in odd rows, the central term, T(n,(n+1)/2) = A058962((n-1)/2); (b) in even rows, the negation of the average of the two central terms, -(T(2n,n) + T(2n,+1))/2 = -A018215(m/2). The absolute values of the alternating row sums give the plain row means, Sum_{k=1..n} T(n,k)/n; the alternating sign row means are (-2)^(n-1). - Gregory Gerard Wojnar, Feb 10 2024

			Triangle begins:
   1;
   2,   6;
   4,  12,  20;
   8,  24,  40,  56;
  16,  48,  80, 112, 144;
  32,  96, 160, 224, 288, 352;
  64, 192, 320, 448, 576, 704, 832;

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A118413, A117303, A054582.

a118416 n k = a118416_tabl !! (n-1) !! (k-1)
a118416_row 1 = [1]
a118416_row n = (map (* 2) $ a118416_row (n-1)) ++ [a014480 (n-1)]
a118416_tabl = map a118416_row [1..]
-- Reinhard Zumkeller, Jan 22 2012

A118416 := proc(n,k) 2^(n-1)*(2*k-1) ; end proc: # R. J. Mathar, Sep 04 2011

Flatten[Table[(2k-1)2^(n-1),{n,10},{k,n}]] (* Harvey P. Dale, Aug 26 2014 *)

from math import isqrt
def A118416(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(1-a)+(n<<1)-1<Chai Wah Wu, Jun 20 2025

T(n,k) = 2*T(n-1,k), 1 <= k < n; T(n,n) = A014480(n-1).

A253887 Row index of n in A191450: a(3n) = 2n, a(3n+1) = 2n+1, a(3n+2) = a(n+1).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 22 2015

nonn

a(n) gives the row index of n in square array A191450, or equally, the column index of n in A254051.

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

Odd bisection of A126760.
Cf. A254046 (the corresponding column index).
Cf. A003602, A032766, A064216, A253786, A253893, A249746.
Cf. A191450, A254051, A254047, A254052, A254104.

def a(n):
    if n%3==0: return 2*n//3
    elif n%3==1: return 2*(n - 1)//3 + 1
    else: return a((n - 2)//3 + 1)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

a(3n) = 2n, a(3n+1) = 2n+1, a(3n+2) = a(n+1).
a(n) = A126760(2n-1).
a(n) = A249746(A003602(A064216(n))). - Antti Karttunen, Feb 04 2015

A209268 Inverse permutation A054582.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 7, 15, 9, 21, 8, 28, 14, 36, 11, 45, 20, 55, 13, 66, 27, 78, 12, 91, 35, 105, 19, 120, 44, 136, 16, 153, 54, 171, 26, 190, 65, 210, 18, 231, 77, 253, 34, 276, 90, 300, 17, 325, 104, 351, 43, 378, 119, 406, 25, 435, 135, 465, 53, 496, 152

Offset: 1

Boris Putievskiy, Jan 15 2013

nonn

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

			The start of the sequence for n = 1..32 as table, distributed by exponent of highest power of 2 dividing n:
   |   Exponent of highest power of 2 dividing n
n  |--------------------------------------------------
   |    0      1      2       3      4         5    ...
------------------------------------------------------
1  |....1
2  |...........2
3  |....3
4  |..................4
5  |....6
6  |...........5
7  |...10
8  |..........................7
9  |...15
10 |...........9
11 |...21
12 |..................8
13 |...28
14 |..........14
15 |...36
16 |................................11
17 |...45
18 |..........20
19 |...55
20 |.................13
21 |...66
22 |..........27
23 |...78
24 |................................12
25 |...91
26 |..........35
27 |..105
28 |.................19
29 |..120
30 |..........44
31 |..136
32 |.........................................16
. . .
Let r_c be number row inside the column number c.
r_c = (n+2^c)/2^(c+1).
The column number 0 contains numbers r_0*(r_0+1)/2,     A000217,
The column number 1 contains numbers r_1*(r_1+3)/2,     A000096,
The column number 2 contains numbers r_2*(r_2+5)/2 + 1, A034856,
The column number 3 contains numbers r_3*(r_3+7)/2 + 3, A055998,
The column number 4 contains numbers r_4*(r_4+9)/2 + 6, A046691.

Boris Putievskiy, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
R. J. Mathar, oeisPy
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A054582, A003602, A007814, A000217, A000096, A034856, A055998, A046691, A014132.

a[n_] := (v = IntegerExponent[n, 2]; (1/2)*(((1/2)*(n/2^v + 1) + v)^2 + (1/2)*(n/2^v + 1) - v)); Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jan 15 2013, from 1st formula *)

f = open("result.csv", "w")
def A007814(n):
### author        Richard J. Mathar 2010-09-06 (Start)
### http://oeis.org/wiki/User:R._J._Mathar/oeisPy/oeisPy/oeis_bulk.py
        a = 0
        nshft = n
        while (nshft %2 == 0):
                a += 1
                nshft >>= 1
        return a
###(End)
for  n in range(1,10001):
     x = A007814(n)
     y = (n+2**x)/2**(x+1)
     m = ((x+y)**2-x+y)/2
     f.write('%d;%d;%d;%d;\n' % (n, x, y, m))
f.close()

a(n) = (((A003602)+A007814(n))^2 - A007814(n) + A003602(n))/2.

a(n) = ((x+y)^2-x+y)/2, where x = max {k: 2^k | n}, y = (n+2^x)/2^(x+1).

cp's OEIS Frontend

A091067 Numbers whose odd part is of the form 4k+3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

PARI

Formula

A078719 Number of odd terms among n, f(n), f(f(n)), ...., 1 for the Collatz function (that is, until reaching "1" for the first time), or -1 if 1 is never reached.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A035528 Euler transform of A027656(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A135764 Distribute the natural numbers in columns based on the occurrence of "2" in each prime factorization; square array A(row,col) = 2^(row-1) * ((2*col)-1), read by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

Extensions

A103391 "Even" fractal sequence for the natural numbers: Deleting every even-indexed term results in the same sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A113415 Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

A118416 Triangle read by rows: T(n,k) = (2k-1)2^(n-1), 0 < k <= n.