A017101 -id:A017101 - cp's OEIS frontend

A047398 Numbers that are congruent to {3, 6} mod 8.

3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, 46, 51, 54, 59, 62, 67, 70, 75, 78, 83, 86, 91, 94, 99, 102, 107, 110, 115, 118, 123, 126, 131, 134, 139, 142, 147, 150, 155, 158, 163, 166, 171, 174, 179, 182, 187, 190, 195, 198, 203, 206, 211, 214, 219, 222, 227, 230

Offset: 1

N. J. A. Sloane

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

Union of A017101 and A017137.

Cf. A047461, A047452, A047470, A047524, A047535, A047615, A047617.

A047398:=n->4*n-(3+(-1)^n)/2: seq(A047398(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2017

Flatten[# + {3, 6} & /@ (8 Range[0, 28])] (* Arkadiusz Wesolowski, Sep 25 2012 *)
LinearRecurrence[{1,1,-1},{3,6,11},60] (* Harvey P. Dale, Oct 26 2020 *)

makelist(4*n + mod(n, 2) - 2, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */

def A047398(n): return ((n<<2)|(n&1))-2 # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Dec 05 2011: (Start)
a(n) = 4*n - (3 + (-1)^n)/2.
G.f.: x*(3+3*x+2*x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), n > 3.
a(n) = 4*n + (n mod 2) - 2.
a(n) = A047470(n) + 3.
a(2*n) = A017137(n-1).
a(2*n-1) = A017101(n-1).
E.g.f.: ((8*x - 3)*exp(x) - exp(-x) + 4)/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 + log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A047471 Numbers that are congruent to {1, 3} mod 8.

Original entry on oeis.org

1, 3, 9, 11, 17, 19, 25, 27, 33, 35, 41, 43, 49, 51, 57, 59, 65, 67, 73, 75, 81, 83, 89, 91, 97, 99, 105, 107, 113, 115, 121, 123, 129, 131, 137, 139, 145, 147, 153, 155, 161, 163, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233

Offset: 1

N. J. A. Sloane

			For n=2, a(2) = 8*2-1-12 = 3;
For n=3, a(3) = 8*3-3-12 = 9;
For n=4, a(4) = 8*4-9-12 = 11. - _Vincenzo Librandi_, Aug 06 2010

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

Union of A017077 and A017101.

Cf. A033200 (primes).

a047471 n = a047471_list !! (n-1)
a047471_list = [n | n <- [1..], mod n 8 `elem` [1,3]]
-- Reinhard Zumkeller, Dec 29 2012

[4*(n-1)-(-1)^n : n in [1..80]]; // Wesley Ivan Hurt, Apr 28 2017

A047471:=n->4*n - 4 - (-1)^n; seq(A047471(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2014

Table[4 n - 4 - (-1)^n, {n, 100}] (* Wesley Ivan Hurt, Jan 30 2014 *)
#+{1,3}&/@(8*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,3,9},60] (* Harvey P. Dale, Jan 05 2017 *)

G.f.: x*(1+2*x+5*x^2)/((1+x)*(1-x)^2). - Paul Barry, Apr 10 2008
a(n) = 4*(n-1)-(-1)^n. - Rolf Pleisch, Aug 04 2009
a(n) = 8*n-a(n-1)-12, with a(1)=1. - Vincenzo Librandi, Aug 06 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + sqrt(2)*log(sqrt(2)+1)/4. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A289840 Complex cross sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 11, 19, 27, 35, 67, 83, 99, 115, 163, 179, 195, 211, 275, 323, 355, 387, 467, 483, 499, 515, 579, 627, 675, 707, 787, 803, 819, 835, 899, 947, 995, 1027, 1107, 1123, 1139, 1155, 1219, 1267, 1315, 1347, 1427, 1443, 1459, 1475, 1539, 1587, 1635, 1667, 1747, 1763, 1779, 1795, 1859, 1907, 1955, 1987, 2067

Offset: 0

Omar E. Pol, Jul 14 2017

The sequence arises from a "hybrid" cellular automaton on the infinite square grid, which consist of two successive generations using toothpicks of length 2 (cf. A139250) followed by two successive generations using the rules of the D-toothpick sequence A220500.
In other words (and more precisely) we have that:
1) If n is congruent to 1 or 2 mod 4 (cf. A042963), for example: 1, 2, 5, 6, 9, 10, ..., the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.
2) If n is a positive integer of the form 4*k-1 (cf. A004767), for example: 3, 7, 11, 15, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) and eventually D-toothpicks of length sqrt(2)/2, in both cases the D-toothpicks are connected to the structure by their endpoints, in the same way as in the even-indexed stages of A220500.
3) If n is a positive multiple of 4 (cf. A008586) the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the odd-indexed stages of A220500.
a(n) is the total number of elements in the structure after n generations.
A289841 (the first differences) gives the number of elements added at n-th stage.
Note that after 19 generations the structure is a 72-gon which essentially looks like a diamond (as a square that has been rotated 45 degrees).
The surprising fact is that from n = 20 up to 27 the structure is gradually transformed into a square cross.
The diamond mentioned above can be interpreted as the center of the cross. The diamond has an area equal to 384 and it contains 222 polygonal regions (or enclosures) of 11 distinct shapes. Missing two heptagonal shapes which are in the arms of the square cross only.
In total the complex cross contains 13 distinct shapes of polygonal regions. There are ten polygonal shapes that have an infinite number of copies. On the other hand, three of these polygonal shapes have a finite number of copies because they are in the center of the cross only. For example: there are only four copies of the concave 14-gon, which is also the largest polygon in the structure.
For n => 27 the shape of the square cross remains forever because its four arms grow indefinitely.
Every arm has a minimum width equal to 8, and a maximum width equal to 12.
Every arm also has a periodic structure which can be dissected in infinitely many clusters of area equal to 64. Every cluster is a 30-gon that contains 40 polygonal regions of nine distinct shapes.
If n is a number of the form 8*k-3 (cf. A017101) and greater than 19, for example: 27, 35, 43, 51, ..., then at n-th stage a new cluster is finished in every arm of the cross.
The behavior is similar to A290220 and A294020 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. - Omar E. Pol, Oct 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A004767, A008586, A017101, A042963, A139250, A220500, A289841, A290220 (a simpler cross), A294020.

concat(0, Vec(x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4))+ O(x^50))) \\ Colin Barker, Nov 12 2017

From Colin Barker, Nov 11 2017: (Start)
G.f.: x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = a(n-1) + a(n-8) - a(n-9) for n>19.
(End)

A290220 Narrow cross sequence (see Comments lines for definition).

Original entry on oeis.org

0, 2, 6, 10, 18, 26, 34, 42, 58, 70, 78, 94, 106, 114, 130, 142, 150, 166, 178, 186, 202, 214, 222, 238, 250, 258, 274, 286, 294, 310, 322, 330, 346, 358, 366, 382, 394, 402, 418, 430, 438, 454, 466, 474, 490, 502, 510, 526, 538, 546, 562, 574, 582, 598, 610, 618, 634, 646, 654, 670, 682, 690, 706, 718, 726, 742, 754

Offset: 0

Omar E. Pol, Jul 24 2017

The sequence arises from a "hybrid" cellular automaton, which consist essentially in two successive generations using the rules of the D-toothpick sequence A194270 followed by one generation using toothpicks of length 2.
On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.
For the next stages we have the following rules:
1) At stage 1 we place two D-toothpicks connected by their endpoints on the same diagonal.
2) If n is a number of the form 3*k + 2 (cf. A016789), for example: 2, 5, 8, 11, 14, ..., the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the even-indexed stages of A194270.
3) If n is a positive multiple of 3 (cf. A008585) the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.
4) If n is a number of the form 3*k + 1 (cf. A016777) and > 1, for example: 4, 7, 10, 13, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) connected to the structure by their endpoints, in the same way as in the odd-indexed stages of A194270.
a(n) is the total number of elements in the structure after n generations.
A290221 (the first differences) gives the number of elements added at n-th stage.
The surprising fact is that from n = 7 up to 9 the structure is gradually transformed into a square cross.
For n => 9 the shape of the square cross remains forever because its four arms grow indefinitely in the directions North, East, West and South.
Every arm has a width equal to 4.
Every arm also has a periodic structure which can be dissected in infinitely many clusters.
In total, the narrow cross contains five distinct shapes of polygonal regions. There are three polygonal shapes that have an infinite number of copies. On the other hand, two polygonal shapes have a finite number of copies because they are in the center of the cross only. they are the heptagon and the hexagon of area 5.
The structure looks like a square cross but it's simpler than the structure of the complex cross described in A289840.
The behavior is similar to A289840 and A294020 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. - Omar E. Pol, Oct 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A004767, A008586, A017101, A042963, A139250, A220500, A289840, A290221, A294020.

LinearRecurrence[{1, 0, 1, -1}, {0, 2, 6, 10, 18, 26, 34, 42, 58, 70}, 100] (* Paolo Xausa, Aug 27 2024 *)

concat(0, Vec(2*x*(1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^5 + 4*x^7 + 2*x^8) / ((1 - x)^2*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Nov 12 2017

From Colin Barker, Nov 11 2017: (Start)
G.f.: 2*x*(1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^5 + 4*x^7 + 2*x^8) / ((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>9. [Corrected by Paolo Xausa, Aug 27 2024]
(End)

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 7, 11, 9, 6, 15, 23, 19, 13, 8, 31, 47, 39, 27, 17, 10, 63, 95, 79, 55, 35, 21, 12, 127, 191, 159, 111, 71, 43, 25, 14, 255, 383, 319, 223, 143, 87, 51, 29, 16, 511, 767, 639, 447, 287, 175, 103, 59, 33, 18, 1023, 1535, 1279, 895, 575, 351, 207, 119

Offset: 0

Antti Karttunen, Sep 12 2002

From Philippe Deléham, Feb 19 2014: (Start)
A(0,k)  = 2*k = A005843(k),
A(1,k)  = 4*k + 1 = A016813(k),
A(2,k)  = 8*k + 3 = A017101(k),
A(n,0)  = A000225(n),
A(n,1)  = A153893(n),
A(n,2)  = A153894(n),
A(n,3)  = A086224(n),
A(n,4)  = A052996(n+2),
A(n,5)  = A086225(n),
A(n,6)  = A198274(n),
A(n,7)  = A238087(n),
A(n,8)  = A198275(n),
A(n,9)  = A198276(n),
A(n,10) = A171389(n). (End)
A permutation of the nonnegative integers. - Alzhekeyev Ascar M, Jun 05 2016
The values in array row n, when expressed in binary, have n trailing 1-bits. - Ruud H.G. van Tol, Mar 18 2025

			The array A begins:
   0    2    4    6    8   10   12   14   16   18 ...
   1    5    9   13   17   21   25   29   33   37 ...
   3   11   19   27   35   43   51   59   67   75 ...
   7   23   39   55   71   87  103  119  135  151 ...
  15   47   79  111  143  175  207  239  271  303 ...
  31   95  159  223  287  351  415  479  543  607 ...
  ... - _Philippe Deléham_, Feb 19 2014
From _Wolfdieter Lang_, Jan 31 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    0
   1:    1    2
   2:    3    5    4
   3:    7   11    9   6
   4:   15   23   19  13   8
   5    31   47   39  27  17  10
   6:   63   95   79  55  35  21  12
   7:  127  191  159 111  71  43  25  14
   8:  255  383  319 223 143  87  51  29 16
   9:  511  767  639 447 287 175 103  59 33 18
  10: 1023 1535 1279 895 575 351 207 119 67 37 20
  ...
T(3, 1) = 2^2*(2*1+1) - 1 = 12 - 1 = 11.  (End)

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A075301. Transpose: A075302. The X-projection is given by A007814(n+1) and the Y-projection A025480.

Cf. A002262, A025581, A054582, A241957.

A075300bi := (x,y) -> (2^x * (2*y + 1))-1;
A075300 := n -> A075300bi(A025581(n), A002262(n));
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);

Table[(2^# (2 k + 1)) - 1 &[m - k], {m, 0, 10}, {k, 0, m}] (* Michael De Vlieger, Jun 05 2016 *)

From Wolfdieter Lang, Jan 31 2019: (Start)
Array A(n, k) = 2^n*(2*k+1) - 1, for n >= 0 and m >= 0.
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1) - 1, n >= 0, k=0..n.
See also A054582 after subtracting 1. (End)
From Ruud H.G. van Tol, Mar 17 2025: (Start)
A(0, k) is even. For n > 0, A(n, k) is odd and (3 * A(n, k) + 1) / 2 = A(n-1, 3*k+1).
A(n, k) = 2^n - 1 (mod 2^(n+1)) (equivalent to the comment about trailing 1-bits). (End)

A106839 Numbers congruent to 11 mod 16.

Original entry on oeis.org

11, 27, 43, 59, 75, 91, 107, 123, 139, 155, 171, 187, 203, 219, 235, 251, 267, 283, 299, 315, 331, 347, 363, 379, 395, 411, 427, 443, 459, 475, 491, 507, 523, 539, 555, 571, 587, 603, 619, 635, 651, 667, 683, 699, 715, 731, 747, 763, 779, 795, 811, 827, 843

Offset: 0

Ralf Stephan, May 03 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Differs from A044072.

Cf. A008598, A017101, A051062, A119413.

[[ n : n in [1..1000] | n mod 16 eq 11]]; // Vincenzo Librandi, Oct 10 2011

Range[11, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
LinearRecurrence[{2,-1},{11,27},60] (* Harvey P. Dale, Aug 12 2021 *)

a(n)=16*n+11 \\ Charles R Greathouse IV, Oct 16 2015

def a(n): return 11 + 16*n
print([a(n) for n in range(53)]) # Michael S. Branicky, Nov 27 2021

G.f.: x*(11+5*x)/(x-1)^2. - R. J. Mathar, Oct 08 2011
From Vincenzo Librandi, Oct 10 2011: (Start)
a(n) = 11 + 16*n.
a(n) = 32*n - a(n-1) + 6, a(0)=11. (End)
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(11 + 16*x).
a(n) = 2*a(n-1) - a(n-2).
a(n) = A017101(2*n+1). (End)

A047457 Numbers that are congruent to {3, 4} mod 8.

Original entry on oeis.org

3, 4, 11, 12, 19, 20, 27, 28, 35, 36, 43, 44, 51, 52, 59, 60, 67, 68, 75, 76, 83, 84, 91, 92, 99, 100, 107, 108, 115, 116, 123, 124, 131, 132, 139, 140, 147, 148, 155, 156, 163, 164, 171, 172, 179, 180, 187, 188, 195, 196, 203, 204, 211, 212, 219, 220, 227

Offset: 1

N. J. A. Sloane

Union of A017101 and A017113. - Michel Marcus, Feb 25 2014

Numbers whose binary reflected Gray code (A014550) has a single trailing zero. - Amiram Eldar, May 17 2021

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

Cf. A000120, A014550, A017101, A017113, A047456 (superset), A063787.

A047457:=n->(-5 - 3*(-1)^n + 8*n)/2; seq(A047457(n), n=1..100); # Wesley Ivan Hurt, Mar 04 2014

Table[(-5 - 3*(-1)^n + 8*n)/2, {n, 100}] (* Wesley Ivan Hurt, Mar 04 2014 *)
Flatten[Table[8n + {3, 4}, {n, 0, 29}]] (* Alonso del Arte, Mar 04 2014 *)

a(n) = 8*n - a(n-1) - 9 (with a(1) = 3). - Vincenzo Librandi, Aug 06 2010
G.f.: x*(3+x+4*x^2)/((1-x)^2*(1+x)). - Colin Barker, May 13 2012
a(n) = (-5 - 3*(-1)^n + 8*n)/2. - Colin Barker, May 14 2012
A000120(a(n)-1) = A000120(a(n)+1) = A063787(n). - Ilya Lopatin and Juri-Stepan Gerasimov, Feb 25 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/16 + log(2)/4 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A004014 Norms of vectors in the b.c.c. lattice.

Original entry on oeis.org

0, 3, 4, 8, 11, 12, 16, 19, 20, 24, 27, 32, 35, 36, 40, 43, 44, 48, 51, 52, 56, 59, 64, 67, 68, 72, 75, 76, 80, 83, 84, 88, 91, 96, 99, 100, 104, 107, 108, 115, 116, 120, 123, 128, 131, 132, 136, 139, 140, 144, 147, 148, 152, 155, 160, 163, 164, 168

Offset: 0

N. J. A. Sloane

Integers such that A004013(n) is nonzero. - Michael Somos, Jul 28 2014
A subsequence of A047458. The complement seems to be 4*A004215. - Andrey Zabolotskiy, Nov 11 2021
From Mohammed Yaseen, Nov 06 2022: (Start)
These are numbers of the form x^2+y^2+z^2 where x, y and z are either all even (including zero) or all odd.
The selection rule for the planes with Miller indices (hkl) to undergo X-ray diffraction in an f.c.c. lattice is h^2+k^2+l^2 = N where N is a term of this sequence. See A000378 for simple cubic lattice. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116. (Chapter 4 section 6.7)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to b.c.c. lattice

Cf. A004013, A047458, A004215.

Union of A034045 and A017101. - Mohammed Yaseen, Nov 06 2022

f:= JacobiTheta2(0,z^4)^3+JacobiTheta3(0,z^4)^3:
S:= series(f,z,1001):
select(t -> coeff(S,z,t) <> 0, [$0..1000]); # Robert Israel, Oct 18 2015

f = EllipticTheta[2, 0, z^4]^3 + EllipticTheta[3, 0, z^4]^3; S = f + O[z]^200; Flatten[Position[CoefficientList[S, z], ?Positive] - 1] (* _Jean-François Alcover, Oct 23 2016, after Robert Israel *)

More terms from Sean A. Irvine, Oct 17 2015

A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 9, 15, 23, 19, 13, 31, 47, 39, 27, 17, 63, 95, 79, 55, 35, 21, 127, 191, 159, 111, 71, 43, 25, 255, 383, 319, 223, 143, 87, 51, 29, 511, 767, 639, 447, 287, 175, 103, 59, 33, 1023, 1535, 1279, 895, 575, 351, 207, 119, 67, 37

Offset: 1

L. Edson Jeffery, Aug 09 2014

The sequence is a permutation of the odd natural numbers, since A(n,k) = 2*A054582(n-1,k-1) - 1 and A054582 is a permutation of the natural numbers.
For j a natural number, 2*j - 1 appears in row A001511(j) of A.
This is the square array A075300 with the first row omitted. - Peter Bala, Feb 07 2017

			Array begins:
.      1     5     9    13    17     21     25     29     33     37
.      3    11    19    27    35     43     51     59     67     75
.      7    23    39    55    71     87    103    119    135    151
.     15    47    79   111   143    175    207    239    271    303
.     31    95   159   223   287    351    415    479    543    607
.     63   191   319   447   575    703    831    959   1087   1215
.    127   383   639   895  1151   1407   1663   1919   2175   2431
.    255   767  1279  1791  2303   2815   3327   3839   4351   4863
.    511  1535  2559  3583  4607   5631   6655   7679   8703   9727
.   1023  3071  5119  7167  9215  11263  13311  15359  17407  19455

Vincenzo Librandi, Rows n = 0..50, flattened
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A016813, A017101 (rows 1 and 2).
Cf. A000225, A083329, A153894, A086224, A052996, etc. (columns 1-5).
Cf. A005408 (odd natural numbers), A054582.
Cf. A075300.

(* Array: *)
Grid[Table[2^n*(2*k - 1) - 1, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[2^(n - k + 1)*(2*k - 1) - 1, {n, 10}, {k, n}]]

A(n,k) = 2*A054582(n-1,k-1) - 1.

A365885 Starts of run of 3 consecutive integers that are terms of A365883.

Original entry on oeis.org

228123, 903123, 1121875, 2253123, 2928123, 3146875, 3821875, 4278123, 5846875, 6303123, 6978123, 7196875, 7871875, 9003123, 9221875, 9896875, 10353123, 11028123, 11246875, 12378123, 13053123, 13271875, 13946875, 14403123, 15971875, 16428123, 17103123, 17321875

Offset: 1

Amiram Eldar, Sep 22 2023

Numbers of the form 4*k+2 are not terms of A365883. Therefore there are no runs of 4 or more consecutive integers.
Since the middle integer in each triple is not divisible by 8, all the terms of this sequence are of the form 8*k+3.
The numbers of terms not exceeding 10^k, for k = 6, 7, ..., are 2, 16, 158, 1585, 15853, 158540, ... . Apparently, the asymptotic density of this sequence exists and equals 1.585...*10^(-6).

			228123 = 3^3 * 7 * 17 * 71 is a term since its least prime factor, 3, is equal to its exponent, the least prime factor of 228123 = 2^2 * 13 * 41 * 107, 2, is equal to its exponent, and the least prime factor of 228125 = 5^5 * 73, 5, is also equal to its exponent.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A017101, A365883, A365884 and A365891.

q[n_] := Equal @@ FactorInteger[n][[1]]; Select[8*Range[125000] + 3, AllTrue[# + {0, 1, 2}, q] &]

is(n) = #Set(factor(n)[1,]) == 1;
lista(kmax) = forstep(k = 3, kmax, 8, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

cp's OEIS Frontend

A047398 Numbers that are congruent to {3, 6} mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Python

Formula

A047471 Numbers that are congruent to {1, 3} mod 8.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

Extensions

A289840 Complex cross sequence (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A290220 Narrow cross sequence (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n*(2*k+1) - 1 with n,k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A106839 Numbers congruent to 11 mod 16.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A047457 Numbers that are congruent to {3, 4} mod 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.

A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.