A010693 -id:A010693 - cp's OEIS frontend

A104772 If n<=2 then n else (if n is odd then 2a(n+1) else pq, where n=p+q, p<=q, primes).

1, 2, 8, 4, 18, 9, 30, 15, 42, 21, 70, 35, 66, 33, 78, 39, 130, 65, 102, 51, 114, 57, 190, 95, 138, 69, 230, 115, 322, 161, 174, 87, 186, 93, 310, 155, 434, 217, 222, 111, 370, 185, 246, 123, 258, 129, 430, 215, 282, 141, 470, 235, 658, 329, 318, 159, 530, 265, 742

Offset: 1

Reinhard Zumkeller, Mar 24 2005

nonn

Encoding of positive integers based on the Goldbach conjecture, see A104774 for decoding: A104774(A104772(n))=n;
a(n - n mod 2) = (2^(1 + n mod 2)) * A020481(floor(n/2))*A020482(floor(n/2));
for numbers greater than 4: a(even) = odd and a(odd) = even;
A001222(a(n)) = A010693(n) for n>2;
a(a(n)) = A104773(n).

Index entries for sequences related to Goldbach conjecture

Cf. A001358, A046315, A090967.

For k>1: a(2*k)=A020481(k)*A020482(k) and a(2*k-1)=2*a(2*k).

A176020 Decimal expansion of (3+sqrt(15))/3.

Original entry on oeis.org

2, 2, 9, 0, 9, 9, 4, 4, 4, 8, 7, 3, 5, 8, 0, 5, 6, 2, 8, 3, 9, 3, 0, 8, 8, 4, 6, 6, 5, 9, 4, 1, 3, 3, 2, 0, 3, 6, 1, 0, 9, 7, 3, 9, 0, 1, 7, 6, 3, 8, 6, 3, 6, 0, 8, 8, 6, 2, 5, 2, 4, 5, 8, 8, 7, 0, 4, 4, 9, 4, 3, 6, 3, 9, 7, 8, 9, 9, 3, 0, 1, 1, 1, 7, 3, 0, 9, 5, 7, 9, 2, 2, 8, 6, 2, 2, 4, 5, 0, 5, 9, 7, 2, 1, 0

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(15))/3 is A010693.

a(n) = A020817(n-1) for n > 1; a(1) = 2.

			(3+sqrt(15))/3 = 2.29099444873580562839...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010472 (decimal expansion of sqrt(15)), A176016 (decimal expansion of (3+sqrt(15))/6), A010693 (repeat 2, 3), A020817 (decimal expansion of 1/sqrt(60)).

RealDigits[(3+Sqrt[15])/3,10,120][[1]] (* Harvey P. Dale, May 20 2013 *)

A029863 Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

Original entry on oeis.org

1, 2, 6, 12, 27, 50, 98, 172, 310, 522, 888, 1444, 2357, 3724, 5882, 9072, 13957, 21082, 31732, 47072, 69545, 101540, 147620, 212516, 304631, 433054, 613030, 861616, 1206089, 1677766, 2324844, 3203748, 4398602, 6009390, 8181250

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n where there are 2 kinds of odd parts and 3 kinds of even parts. - Ilya Gutkovskiy, Jan 17 2018

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 27*x^4 + 50*x^5 + 98*x^6 + 172*x^7 + ...

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, p. 16.
Songxin Liang & Jingzhong Zhang, A complete discrimination system for polynomials with complex coefficients and its automatic generation, 1999, p. 2. the sequence Cn is the numbers of cases for root classification of an n-degree polynomial with complex coefficients.
N. J. A. Sloane, Transforms

Cf. A002513, A010693, A085140, A262380.

nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)*(1 - x^k)^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 20 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (eta(x + A)^2 * eta(x^2 + A)), n))};

Euler transform of period 2 sequence [2, 3, ...].
a(n) ~ 5 * exp(sqrt(5*n/3)*Pi) / (48 * n^(3/2)). - Vaclav Kotesovec, Sep 20 2015
G.f.: Product_{k >= 1} 1/(1-x^k)^A010693(k-1). - Georg Fischer, Dec 10 2020

A368227 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006369 map.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 5, 3, 3, 1, 0, 6, 7, 7, 2, 2, 1, 0, 7, 4, 9, 9, 3, 3, 1, 0, 8, 9, 5, 6, 6, 2, 2, 1, 0, 9, 11, 6, 7, 4, 4, 3, 3, 1, 0, 10, 6, 15, 4, 9, 5, 5, 2, 2, 1, 0, 11, 13, 4, 10, 5, 6, 7, 7, 3, 3, 1, 0, 12, 15, 17, 5, 13, 7, 4, 9, 9, 2, 2, 1, 0

Offset: 0

Paolo Xausa, Dec 18 2023

			Array begins:
  [ 0]   0,  0,  0,  0,  0,  0,  0,  0,  0,   0,  0, ... = A000004
  [ 1]   1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1, ... = A000012
  [ 2]   2,  3,  2,  3,  2,  3,  2,  3,  2,   3,  2, ... = A010693
  [ 3]   3,  2,  3,  2,  3,  2,  3,  2,  3,   2,  3, ... = A176059
  [ 4]   4,  5,  7,  9,  6,  4,  5,  7,  9,   6,  4, ... = A094328
  [ 5]   5,  7,  9,  6,  4,  5,  7,  9,  6,   4,  5, ... = A094328 (shifted)
  [ 6]   6,  4,  5,  7,  9,  6,  4,  5,  7,   9,  6, ... = A094328 (shifted)
  [ 7]   7,  9,  6,  4,  5,  7,  9,  6,  4,   5,  7, ... = A094328 (shifted)
  [ 8]   8, 11, 15, 10, 13, 17, 23, 31, 41,  55, 73, ... = A028394
  [ 9]   9,  6,  4,  5,  7,  9,  6,  4,  5,   7,  9, ... = A094328 (shifted)
  [10]  10, 13, 17, 23, 31, 41, 55, 73, 97, 129, 86, ... = A028394 (shifted)
  ...    |   |   |
      A001477|A168222
          A006369

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem

Rows: A000004, A000012, A010693, A028394, A028396, A094328, A094329, A176059, A185589, A185590, A217729, A223083, A223084, A223085.
Columns: A001477, A006369, A168222.
Main diagonal: A368228.
Cf. A368179.

A006369[n_]:=If[Divisible[n,3],2n/3,Round[4n/3]];
A368227list[dmax_]:=With[{a=Reverse[Table[NestList[A006369,n-1,dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368227list[15] (* Generates 15 antidiagonals *)

A242112 a(n) = floor((2*n+6)/(5-(-1)^n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 5, 4, 6, 4, 7, 5, 8, 6, 9, 6, 10, 7, 11, 8, 12, 8, 13, 9, 14, 10, 15, 10, 16, 11, 17, 12, 18, 12, 19, 13, 20, 14, 21, 14, 22, 15, 23, 16, 24, 16, 25, 17, 26, 18, 27, 18, 28, 19, 29, 20, 30, 20, 31, 21, 32, 22, 33, 22, 34, 23, 35, 24, 36

Offset: 0

Wesley Ivan Hurt, Aug 21 2014

Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1,0,-1).

Cf. A010693, A093718.

[Floor((2*n+6)/(5-(-1)^n)) : n in [0..100]];

[IsEven(n) select 1+n/2 else 1+Floor(n/3): n in [0..80]]; // Bruno Berselli, Aug 22 2014

A242112:=n->floor((2*n+6)/(5-(-1)^n)): seq(A242112(n), n=0..100);

Table[Floor[(2 n + 6)/(5 - (-1)^n)], {n, 0, 100}]
LinearRecurrence[{0,1,0,0,0,1,0,-1},{1,1,2,2,3,2,4,3},80] (* Harvey P. Dale, Oct 24 2017 *)

a(n) = a(n-2) + a(n-6) - a(n-8).
a(n) = ( n+3 - A093718(n) ) / A010693(n).
From Robert Israel, Aug 22 2014: (Start)
a(n) = sqrt(3)/18*(sin(2*n*Pi/3)+sin(n*Pi/3)) + 1/6*(cos(2*n*Pi/3)-cos(n*Pi/3)) + (-1)^n*(2+n)/12 + 5*(n+2)/12.
G.f.: (1 + x + x^2 + x^3 + x^4)/(1 - x^2 - x^6 + x^8). (End)
a(n) = 1 + n/2 if n is even, otherwise a(n) = 1 + floor(n/3). - Bruno Berselli, Aug 22 2014

A075328 Difference between n-th pair in A075325.

Original entry on oeis.org

2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65, 67, 70, 72, 75, 77, 80, 82, 85, 87, 90, 92, 95, 97, 100, 102, 105, 107, 110, 112, 115, 117, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 150, 152, 155

Offset: 0

Amarnath Murthy, Sep 18 2002

nonn

Empirically the partial sums of A010693 (i.e., 2, 3 repeated). - Sean A. Irvine, Jul 12 2022

Cf. A075325, A075326, A075327.

Cf. A010693, A047215.

More terms from David Wasserman, Jan 16 2005

A135537 Period 4: repeat [7, 5, 2, 4].

Original entry on oeis.org

7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5, 2, 4, 7, 5

Offset: 0

Paul Curtz, Feb 22 2008

Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A010686, A010693, A021408, A135536.

&cat [[7, 5, 2, 4]^^30]; // Wesley Ivan Hurt, Jul 08 2016

seq(op([7, 5, 2, 4]), n=0..50); # Wesley Ivan Hurt, Jul 08 2016

Flatten[Table[{7,5,2,4},{20}]] (* Harvey P. Dale, Jun 22 2011 *)

a(n) = A135536(n) mod 9.
O.g.f.: ((x+5)/(x^2+1)+9*(1-x))/2. a(n) = ((-1)^floor(n/2) * A010686(n+1) + 9)/2. - R. J. Mathar, Feb 23 2008
From Wesley Ivan Hurt, Jul 08 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n>2, a(n) = a(n-4) for n>3.
a(n) = A021408(n) for n>1. (End)

cp's OEIS Frontend

A104772 If n<=2 then n else (if n is odd then 2*a(n+1) else p*q, where n=p+q, p<=q, primes).

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A176020 Decimal expansion of (3+sqrt(15))/3.

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A029863 Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

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A368227 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006369 map.

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A242112 a(n) = floor((2*n+6)/(5-(-1)^n)).

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A075328 Difference between n-th pair in A075325.

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A135537 Period 4: repeat [7, 5, 2, 4].

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A104772 If n<=2 then n else (if n is odd then 2a(n+1) else pq, where n=p+q, p<=q, primes).