A047241 -id:A047241 - cp's OEIS frontend

A047255 Numbers that are congruent to {1, 2, 3, 5} mod 6.

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101, 103, 104

Offset: 1

N. J. A. Sloane

Each element is coprime to preceding two elements. - Amarnath Murthy, Jun 12 2001

The sequence is the interleaving of A047241 with A016789. - Guenther Schrack, Feb 16 2019

			After 21 and 23 the next term is 25 as 24 has a common divisor with 21.

Guenther Schrack, Table of n, a(n) for n = 1..10012
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A007310, A016789, A047228, A047237, A047241, A047251, A047261, A047273, A062062, A062063.

Complement: A047233

a047255 n = a047255_list !! (n-1)
a047255_list = 1 : 2 : 3 : 5 : map (+ 6) a047255_list
-- Reinhard Zumkeller, Jan 17 2014

[n : n in [0..100] | n mod 6 in [1, 2, 3, 5]]; // Wesley Ivan Hurt, May 21 2016

A047255:=n->(6*n-4+I^(1-n)+I^(n-1))/4: seq(A047255(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Select[Range[100], MemberQ[{1, 2, 3, 5}, Mod[#, 6]] &]
LinearRecurrence[{2,-2,2,-1},{1,2,3,5},100] (* Harvey P. Dale, May 14 2020 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,2,-2,2]^(n-1)*[1;2;3;5])[1,1] \\ Charles R Greathouse IV, Feb 11 2017

a=(x*(1+x^2+x^3)/((1+x^2)*(1-x)^2)).series(x, 80).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

{k | k == 1, 2, 3, 5 (mod 6)}.
G.f.: x*(1 + x^2 + x^3) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4), for n>4.
a(n) = (6*n - 4 + i^(1-n) + i^(n-1))/4, where i = sqrt(-1).
a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047241(n). (End)
E.g.f.: (2 + sin(x) + (3*x - 2)*exp(x))/2. - Ilya Gutkovskiy, May 21 2016
a(1-n) = - A047251(n). - Wesley Ivan Hurt, May 21 2016
From Guenther Schrack, Feb 16 2019: (Start)
a(n) = (6*n - 4 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=1, a(2)=2, a(3)=3, a(4)=5, for n > 4.
a(n) = A047237(n) + 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 + log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021
a(n) = 2*n - 1 - floor(n/2) + floor(n/4) - floor((n+1)/4). - Ridouane Oudra, Feb 21 2023

More terms from Larry Reeves (larryr(AT)acm.org), Jun 15 2001

A047261 Numbers that are congruent to {2, 4, 5} mod 6.

Original entry on oeis.org

2, 4, 5, 8, 10, 11, 14, 16, 17, 20, 22, 23, 26, 28, 29, 32, 34, 35, 38, 40, 41, 44, 46, 47, 50, 52, 53, 56, 58, 59, 62, 64, 65, 68, 70, 71, 74, 76, 77, 80, 82, 83, 86, 88, 89, 92, 94, 95, 98, 100, 101, 104, 106, 107, 110, 112, 113, 116, 118, 119, 122, 124

Offset: 1

N. J. A. Sloane

If B and C are terms in the sequence then 2*B*C is a term. B (resp. C) is a term iff B (resp. C) mod 6 = 2, 4 or 5. It follows that (2*B*C) mod 6 = (2*(B mod 6)*(C mod 6)) mod 6 = 2 or 4 and therefore 2*B*C is a term. Examples: for B=16 and C=29, 2*16*29 = 928 is a term: (2*B*C) mod 6 = (2*16*29) mod 6 = 4; (2*2*2) mod 6 = 2. - Jerzy R Borysowicz, May 24 2018

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

Cf. A047242 (complement).

Cf. A007310, A047228, A047241, A047273, A056970, A214090.

a047261 n = a047261_list !! n
a047261_list = 2 : 4 : 5 : map (+ 6) a047261_list
-- Reinhard Zumkeller, Feb 19 2013, Jul 06 2012

[n : n in [0..150] | n mod 6 in [2, 4, 5]]; // Wesley Ivan Hurt, Jun 14 2016

A047261:=n->(6*n-1-2*cos(2*n*Pi/3))/3: seq(A047261(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

CoefficientList[Series[(1 + x)*(x^2 + 2)/((1 + x + x^2)*(x - 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Aug 16 2014 *)
Select[ Range@ 125, MemberQ[{2, 4, 5}, Mod[#, 6]] &] (* or *)
LinearRecurrence[{1, 0, 1, -1}, {2, 4, 5, 8}, 62] (* Robert G. Wilson v, Jun 13 2018 *)

G.f.: x*(1+x)*(x^2+2) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
A214090(a(n)) = 1. - Reinhard Zumkeller, Jul 06 2012
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (6*n - 1 - 2*cos(2*n*Pi/3))/3.
a(3k) = 6k-1, a(3k-1) = 6k-2, a(3k-2) = 6k-4. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2+sqrt(3))/(2*sqrt(3)) + log(2)/3. - Amiram Eldar, Dec 16 2021
E.g.f.: (3 + exp(x)*(6*x - 1) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Jul 26 2024

More terms from Wesley Ivan Hurt, Aug 16 2014

A001201 Number of Steiner triple systems (STS's) on 6n+1 or 6n+3 elements.

Original entry on oeis.org

1, 1, 30, 840, 1197504000, 60281712691200, 1348410350618155344199680000

Offset: 0

N. J. A. Sloane

To be precise, for n even, a(n) is the number of STS's on 3n+1 elements, and for n odd, a(n) is the number of STS's on 3n elements. - Franklin T. Adams-Watters, Apr 10 2010

			There are 1197504000 STS's on 13 elements.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 304.
CRC Handbook of Combinatorial Designs, 1996, p. 70.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Petteri Kaski and Patric R. J. Östergård, The Steiner triple systems of order 19, Math. Comp. 73 (2004), 2075-2092.
Index entries for sequences related to Steiner systems

Indexed by A047241. Cf. A030128, A030129, A051390.

A087445 Numbers that are congruent to 1 or 5 mod 12.

Original entry on oeis.org

1, 5, 13, 17, 25, 29, 37, 41, 49, 53, 61, 65, 73, 77, 85, 89, 97, 101, 109, 113, 121, 125, 133, 137, 145, 149, 157, 161, 169, 173, 181, 185, 193, 197, 205, 209, 217, 221, 229, 233, 241, 245, 253, 257, 265, 269, 277, 281, 289, 293, 301, 305, 313, 317, 325, 329

Offset: 1

Paul Barry, Sep 04 2003

From Bob Selcoe, Jun 03 2015: (Start)
For k >= 1: all numbers congruent to A002450(k) mod 2^(2k+1) and A072197(k) mod 4^(k+1) not congruent to 0 mod 3. Equivalently, for k >= 3: all numbers congruent to A096773(k) mod 2^k not congruent to 0 mod 3.
Conjecture: at least one number in this sequence must appear in all Collatz sequences.
(End)
The sequence is composed of all numbers in congruence classes T(n,1) mod 2^(n+k) in A259663 (i.e., T"(1) in array T259663(n,k)) not congruent to 0 mod 3. Therefore the conjecture above is true (see A259663 for additional explanation). - Bob Selcoe, Jul 15 2017
Closure of {1} under the map (x,y)->2x+3y [Klarner-Rado, see Lagarias (2016), p. 755]. - N. J. A. Sloane, Oct 06 2016
The above conjecture is true: this is because even numbers and odd numbers divisible by 3 will lead to the set of odd numbers not divisible by 3. Odd numbers of the form 4k - 1 can also be ignored, as this consists of odd numbers that grow between themselves and the next odd term through Collatz iteration. No infinite sequence of growth between consecutive odd terms is possible, so all numbers of the form 4k - 1 will lead to an odd number that shrinks between itself and the next odd number. All numbers 4k - 1 will lead to a number in 4k - 3, the odd numbers that shrink between themselves and the following odd term. What we are left after that elimination is this sequence. - Aidan Simmons, Feb 25 2019

Jeffrey C. Lagarias, Erdős, Klarner and the 3x+ 1 Problem, Amer. Math. Monthly, Vol. 123, No. 8 (2016), pp. 753-776.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A002450, A047241, A087444, A087446, A096773, A259663.

[k:k in [1..330]| k mod 12 in [1,5]]; // Marius A. Burtea, Feb 08 2020

seq(6*(n-1)-(-1)^n,n=1..100); # Robert Israel, Jun 10 2015

LinearRecurrence[{1,1,-1},{1,5,13},70] (* or *) Rest[CoefficientList[ Series[x (1+4x+7x^2)/((1+x)(1-x)^2),{x,0,70}],x]]  (* Harvey P. Dale, Jun 13 2011 *)

a(n)=(n-1)\2*12 + [5,1][n%2+1] \\ Charles R Greathouse IV, Jun 03 2015

G.f.: x*(1+4*x+7*x^2)/((1+x)*(1-x)^2).
E.g.f.: 6*(x-1)*exp(x) + 7 - exp(-x). - corrected by Robert Israel, Jun 10 2015
a(n) = 6*(n-1) - (-1)^n. - Rolf Pleisch, Aug 04 2009
a(n) = 12*n - a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
a(n) = a(n-1) + a(n-2) - a(n-3), with a(0)=1, a(1)=5, a(2)=13. - Harvey P. Dale, Jun 13 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + log(2 + sqrt(3))/(2*sqrt(3)). - Amiram Eldar, Dec 28 2021

A186422 First differences of A186421.

Original entry on oeis.org

1, 1, -1, 3, -1, 3, -3, 5, -3, 5, -5, 7, -5, 7, -7, 9, -7, 9, -9, 11, -9, 11, -11, 13, -11, 13, -13, 15, -13, 15, -15, 17, -15, 17, -17, 19, -17, 19, -19, 21, -19, 21, -21, 23, -21, 23, -23, 25, -23, 25, -25, 27, -25, 27, -27, 29, -27, 29, -29, 31, -29, 31, -31, 33, -31, 33, -33, 35, -33, 35, -35, 37, -35, 37, -37, 39, -37, 39, -39, 41, -39, 41, -41, 43

Offset: 0

Reinhard Zumkeller, Feb 21 2011

a(n) = A186421(n+1) - A186421(n);
a(2*n) = - A109613(n-1) for n>0; a(2*n+1) = A109613(n);
a(3*k) = A047270(floor((k+1)/2)) * (-1)^(k+1);
a(3*k+1) = A007310(floor((k+2)/2)) * (-1)^k;
a(3*k+2) = A047241(floor((k+3)/2)) * (-1)^(k+1).

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

Cf. A007310, A047241, A047270, A109613, A186421.

a186422 n = a186422_list !! n
a186422_list = zipWith (-) (tail a186421_list) a186421_list

/* By definition: */
A186421:=func;
[A186421(n+1)-A186421(n): n in [0..90]]; // Bruno Berselli, Mar 04 2013

Differences@ CoefficientList[Series[x (1 + 2 x + 2 x^3 + x^4)/((1 + x^2) (x - 1)^2 (1 + x)^2), {x, 0, 84}], x] (* Michael De Vlieger, Oct 02 2017 *)

makelist(-((2*n+1)*(-1)^n-2*%i^(n*(n+1))-3)/4,n,0,83); /* Bruno Berselli, Mar 04 2013 */

G.f.: -(x^4+2*x^3+2*x+1) / ((x-1)*(x+1)^2*(x^2+1)). - Colin Barker, Mar 04 2013
a(n) = -((2*n+1)*(-1)^n-2*i^(n*(n+1))-3)/4, where i=sqrt(-1). [Bruno Berselli, Mar 04 2013]
a(n) = cos((n-1)*Pi)*(2*n+1-2*cos(n*Pi/2)-3*cos(n*Pi)-2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
E.g.f.: (cos(x) + (1 + x)*cosh(x) - sin(x) - (x - 2)*sinh(x))/2. - Stefano Spezia, May 09 2021

A047246 Numbers that are congruent to {0, 1, 2, 3} mod 6.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 14, 15, 18, 19, 20, 21, 24, 25, 26, 27, 30, 31, 32, 33, 36, 37, 38, 39, 42, 43, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 61, 62, 63, 66, 67, 68, 69, 72, 73, 74, 75, 78, 79, 80, 81, 84, 85, 86, 87, 90, 91, 92, 93, 96, 97, 98

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047238 with A047241. - Guenther Schrack, Feb 12 2019

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

Cf. A045331 (primes congruent to {1,2,3} mod 6), A047238, A047241.

Complement: A047257.

Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 2 or n mod 6 = 3); # Muniru A Asiru, Feb 20 2019

a047246 n = a047246_list !! (n-1)
a047246_list = [0..3] ++ map (+ 6) a047246_list
-- Reinhard Zumkeller, Jan 15 2013

[Floor((6/5)*Floor(5*(n-1)/4)) : n in [1..100]]; // Wesley Ivan Hurt, May 21 2016

A047246:=n->(6*n-9-I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/4: seq(A047246(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(6n-9-I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(1+x+x^2+3*x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(1+x+x^2+3*x^3)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x^2*(1+x+x^2+3*x^3) / ((1+x)*(1-x)^2*(1+x^2)). - R. J. Mathar, Oct 08 2011
a(n) = floor((6/5)*floor(5*(n-1)/4)). - Bruno Berselli, May 03 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 9 - i^(2*n) - (1-i)*i^(-n) - (1+i)*i^n)/4 where i=sqrt(-1).
a(2*n) = A047241(n), a(2*n-1) = A047238(n). (End)
E.g.f.: (6 + sin(x) - cos(x) + (3*x - 4)*sinh(x) + (3*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Guenther Schrack, Feb 12 2019: (Start)
a(n) = (6*n - 9 - (-1)^n - 2*(-1)^(n*(n+1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=2, a(4)=3, for n > 4. (End)
Sum_{n>=2} (-1)^n/a(n) = Pi/(6*sqrt(3)) + 2*log(2)/3. - Amiram Eldar, Dec 16 2021
a(n)-a(n-1) = A093148(n-2). - R. J. Mathar, May 01 2024

More terms from Wesley Ivan Hurt, May 21 2016

A258207 Square array: row n gives the numbers remaining after the stage n of Lucky sieve.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 7, 3, 1, 9, 9, 7, 3, 1, 11, 13, 9, 7, 3, 1, 13, 15, 13, 9, 7, 3, 1, 15, 19, 15, 13, 9, 7, 3, 1, 17, 21, 21, 15, 13, 9, 7, 3, 1, 19, 25, 25, 21, 15, 13, 9, 7, 3, 1, 21, 27, 27, 25, 21, 15, 13, 9, 7, 3, 1, 23, 31, 31, 31, 25, 21, 15, 13, 9, 7, 3, 1, 25, 33, 33, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1, 27, 37, 37, 37, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1, 29, 39, 43, 43, 37, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1

Offset: 1

Antti Karttunen, Jul 27 2015

This square array A(row,col) is read by downwards antidiagonals as: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Lucky sieve starts with natural numbers: 1, 2, 3, 4, 5, 6, 7, ... from which at first stage the even numbers are removed, and on each subsequent stage n (n > 1) one sets k =  (these k will form the Lucky numbers) and removes every k-th term (from column positions k, 2k, 3k, etc.) of the preceding row to produce the next row of this array.
On each row n, the first term that differs from the n-th Lucky number (A000959(n)) occurs at the column position A000959(n+1) and that number is A219178(n) when n > 1.

			The top left corner of the array:
1, 3, 5, 7,  9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39
1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57
1, 3, 7, 9, 13, 15, 21, 25, 27, 31, 33, 37, 43, 45, 49, 51, 55, 57, 63, 67
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 45, 49, 51, 55, 63, 67, 69, 73
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 55, 63, 67, 69, 73, 75
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79
...
To get row 2 from row 1, we use the second term of the first row, which is 3, to remove every third term from row 1: 5, 11, 17, ... which leaves 1, 3, 7, 9, 13, ...
To get row 3 from row 2, we use the third term of row 2, which is 7, to remove every seventh term from row 2: 19, 39, ... which then results in the third row.

Cf. A000959 (Lucky numbers), which occur at the main and also any subdiagonal of this array. Also the rows converge towards A000959.
Row 1: A005408. Row 2: A047241. Row 3: A258011.
Transpose: A258208.
Cf. also A219178, A255543, A260717.

(define (A258207 n) (A258207bi (A002260 n) (A004736 n)))
(define (A258207bi row col) ((rowfun_n_for_A000959sieve row) col))
;; Uses definec-macro which can memoize also function-closures:
(definec (rowfun_n_for_A000959sieve n) (if (= 1 n) A005408shifted (let* ((prevrowfun (rowfun_n_for_A000959sieve (- n 1))) (everynth (prevrowfun n))) (compose-funs prevrowfun (nonzero-pos 1 1 (lambda (i) (modulo i everynth)))))))
(define (A005408shifted n) (- (* 2 n) 1))

A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34, 37, 38, 39, 40, 43, 44, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 70, 73, 74, 75, 76, 79, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 97, 98

Offset: 1

N. J. A. Sloane

a(k)^m is a term for k and m in N. - Jerzy R Borysowicz, Apr 18 2023

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

Cf. A008621, A047235, A047241, A047245, A047246, A047247, A093148.

Complement of A047264. Equals A203016 divided by 3.

[n: n in [0..100] | n mod 6 in [1..4]]; // Vincenzo Librandi, Jan 06 2013

A047227:=n->(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A047227(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Complement[Range[100], Flatten[Table[{6n - 1, 6n}, {n, 0, 15}]]] (* Alonso del Arte, Jul 07 2011 *)
Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^(n-1)*[1;2;3;4;7])[1,1] \\ Charles R Greathouse IV, May 03 2023

From Johannes W. Meijer, Jul 07 2011: (Start)
a(n) = floor((n+2)/4) + floor((n+1)/4) + floor(n/4) + 2*floor((n-1)/4) + floor((n+3)/4).
G.f.: x*(1 + x + x^2 + x^3 + 2*x^4)/(x^5 - x^4 - x + 1). (End)
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6n - 5 - i^(2n) + (1+i)*i^(1-n) + (1-i)*i^(n-1))/4 where i=sqrt(-1).
a(2n) = A047235(n), a(2n-1) = A047241(n). (End)
E.g.f.: (4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = A047246(n) + 1.
a(n+2) - a(n+1) = A093148(n) for n>0.
a(1-n) = - A047247(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + 2*log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021

A278492 Square array where row n (n >= 0) gives the numbers remaining after the n-th round of the Flavius sieve, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 3, 1, 5, 7, 7, 3, 1, 6, 9, 9, 7, 3, 1, 7, 11, 13, 13, 7, 3, 1, 8, 13, 15, 15, 13, 7, 3, 1, 9, 15, 19, 19, 19, 13, 7, 3, 1, 10, 17, 21, 25, 25, 19, 13, 7, 3, 1, 11, 19, 25, 27, 27, 27, 19, 13, 7, 3, 1, 12, 21, 27, 31, 31, 31, 27, 19, 13, 7, 3, 1, 13, 23, 31, 37, 39, 39, 39, 27, 19, 13, 7, 3, 1

Offset: 0

Antti Karttunen, Nov 23 2016, after David W. Wilson's posting on SeqFan-list Nov 22 2016

The terms of square array A(row,col) are read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left corner of the array:
1, 2, 3,  4,  5,  6,  7,  8,  9, 10 (row 0: start from A000027)
1, 3, 5,  7,  9, 11, 13, 15, 17, 19 (after the 1st round, A005408 remain)
1, 3, 7,  9, 13, 15, 19, 21, 25, 27 (after the 2nd, A047241)
1, 3, 7, 13, 15, 19, 25, 27, 31, 37
1, 3, 7, 13, 19, 25, 27, 31, 39, 43
1, 3, 7, 13, 19, 27, 31, 39, 43, 49
1, 3, 7, 13, 19, 27, 39, 43, 49, 61
1, 3, 7, 13, 19, 27, 39, 49, 61, 63
1, 3, 7, 13, 19, 27, 39, 49, 63, 67
1, 3, 7, 13, 19, 27, 39, 49, 63, 79

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of the array
D. Wilson et al., Interesting sequence, SeqFan list, Nov. 2016
Index entries for sequences generated by sieves
Index entries for sequences related to the Josephus Problem

One more than A278482.
Transpose: A278493.
Rows: A000027, A005408, A047241, A056530, A056531.
Main diagonal: A000960.
Cf. A278507 (the numbers removed at each round).
Similarly constructed arrays for other sieves: A258207, A260717.

(define (A278492 n) (A278492bi (A002262 n) (A025581 n)))
(define (A278492bi row col) (+ 1 (A278482bi row col)))

A(n,k) = 1 + A278482(n,k).

A056531 Sequence remaining after a fourth round of Flavius Josephus sieve; remove every fifth term of A056530.

Original entry on oeis.org

1, 3, 7, 13, 19, 25, 27, 31, 39, 43, 49, 51, 61, 63, 67, 73, 79, 85, 87, 91, 99, 103, 109, 111, 121, 123, 127, 133, 139, 145, 147, 151, 159, 163, 169, 171, 181, 183, 187, 193, 199, 205, 207, 211, 219, 223, 229, 231, 241, 243, 247, 253, 259, 265, 267, 271, 279

Offset: 1

Henry Bottomley, Jun 19 2000

Numbers {1, 3, 7, 13, 19, 25, 27, 31, 39, 43, 49, 51} mod 60.

Index entries for sequences related to the Josephus Problem
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Compare A000027 for 0 rounds of sieve, A005408 after 1 round of sieve, A047241 after 2 rounds, A056530 after 3 rounds, A056531 after 4 rounds, A000960 after all rounds.

After n rounds the remaining sequence comprises A002944(n) numbers mod A003418(n+1), i.e. 1/(n+1) of them.

LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,1,-1},{1,3,7,13,19,25,27,31,39,43,49,51,61},60] (* Harvey P. Dale, Mar 11 2019 *)

From Chai Wah Wu, Jul 24 2016: (Start)
a(n) = a(n-1) + a(n-12) - a(n-13) for n > 13.
G.f.: x*(9*x^12 + 2*x^11 + 6*x^10 + 4*x^9 + 8*x^8 + 4*x^7 + 2*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 4*x^2 + 2*x + 1)/(x^13 - x^12 - x + 1). (End)

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A047261 Numbers that are congruent to {2, 4, 5} mod 6.

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A001201 Number of Steiner triple systems (STS's) on 6n+1 or 6n+3 elements.

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A186422 First differences of A186421.

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A047246 Numbers that are congruent to {0, 1, 2, 3} mod 6.

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