A259748 -id:A259748 - cp's OEIS frontend

A008606 Multiples of 24.

0, 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, 408, 432, 456, 480, 504, 528, 552, 576, 600, 624, 648, 672, 696, 720, 744, 768, 792, 816, 840, 864, 888, 912, 936, 960, 984, 1008, 1032, 1056, 1080, 1104, 1128, 1152, 1176, 1200

Offset: 0

N. J. A. Sloane

If n is a multiple of 24, also the sum of the divisors of n-1 is a multiple of 24. - Vincenzo Librandi, Apr 12 2014

This is the sequence of numbers n such that A259748(n)/n = 11/12. - Danny Rorabaugh, Oct 22 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 336.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008604, A008605.

Range[0, 1500, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[24 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 11 2013 *)

a(n)=24*n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 24*x/(1-x)^2. - Vincenzo Librandi, Jun 11 2013
a(n) = 24*A001477(n) - Danny Rorabaugh, Oct 24 2015
E.g.f.: 24*x*exp(x). - Stefano Spezia, Mar 02 2025

A073762 a(n) = 24*n - 12.

Original entry on oeis.org

12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 852, 876, 900, 924, 948, 972, 996, 1020, 1044, 1068, 1092, 1116, 1140, 1164, 1188, 1212

Offset: 1

Labos Elemer, Aug 08 2002

Previous name: "Smallest unrelated number belonging to a term of this sequence equals 8."
This is also the list of numbers k such that A259748(k)/k = 5/12. - José María Grau Ribas, Jul 12 2015.
Also the total number of line segments creating a stellated octahedron, where the length of each stellated edge equals n-1, and where the octahedron has 12 edges, each fixed at unit length. - Peter M. Chema, Apr 28 2016

			URSet[12] = {8,9,10} so 12 is here.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A016825, A045763, A073758, A073759, A073760, A259748.

[24*n-12: n in [1..60]]; // Vincenzo Librandi, Jun 15 2011

tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Do[s=Min[unr[n]]; If[Equal[s, 8], Print[n]], {n, 1, 1000}]
Range[12, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)

a(n)=24*n-12 \\ Charles R Greathouse IV, Jun 14 2011

x='x+O('x^100); Vec(12*(1+x)/(1-x)^2) \\ Altug Alkan, Oct 22 2015

Min{URS[m]} = 8, where UNR[m] = Complement[RRS[m], Divisors[m]].
a(n) = 24*n - 12. - Max Alekseyev, Mar 03 2007
a(n) = 12*A005408(n-1). - Danny Rorabaugh, Oct 22 2015
G.f.: 12*x*(1 + x)/(1 - x)^2. - Ilya Gutkovskiy, Apr 28 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/48. - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 12*(exp(x)*(2*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A259750 Numbers that are congruent to {14, 22} mod 24.

Original entry on oeis.org

14, 22, 38, 46, 62, 70, 86, 94, 110, 118, 134, 142, 158, 166, 182, 190, 206, 214, 230, 238, 254, 262, 278, 286, 302, 310, 326, 334, 350, 358, 374, 382, 398, 406, 422, 430, 446, 454, 470, 478, 494, 502, 518, 526, 542, 550, 566, 574, 590, 598, 614, 622, 638

Offset: 1

José María Grau Ribas, Jul 04 2015

Original name: Numbers n such that n/A259748(n) = 2.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000914, A168489, A259748.

A[n_] := A[n] = Sum[a b, {a, 1,  n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]/# == 1/2 & ]

vector(100, n, 2*(6*n-(-1)^n)) \\ Altug Alkan, Oct 23 2015

Vec(2*x*(7+4*x+x^2)/((1-x)^2*(1+x)) + O(x^100)) \\ Colin Barker, Aug 26 2016

A259748(a(n))/a(n) = 1/2.
a(n) = 2*A168489(n) - Danny Rorabaugh, Oct 22 2015
From Colin Barker, Aug 26 2016: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: 2*x*(7+4*x+x^2) / ((1-x)^2*(1+x)).
(End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/24 - log(2+sqrt(3))/(4*sqrt(3)). - Amiram Eldar, Dec 31 2021
E.g.f.: 2*(1 + 6*x*exp(x) - exp(-x)). - David Lovler, Sep 06 2022

Better name from Danny Rorabaugh, Oct 22 2015

A259755 Numbers that are congruent to {4, 20} mod 24.

Original entry on oeis.org

4, 20, 28, 44, 52, 68, 76, 92, 100, 116, 124, 140, 148, 164, 172, 188, 196, 212, 220, 236, 244, 260, 268, 284, 292, 308, 316, 332, 340, 356, 364, 380, 388, 404, 412, 428, 436, 452, 460, 476, 484, 500, 508, 524, 532, 548, 556, 572, 580, 596, 604, 620, 628

Offset: 1

José María Grau Ribas, Jul 18 2015

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000914, A007310.

Other sequences of numbers k such that A259748(k)/k equals a constant: A008606, A073762, A259749, A259750, A259751, A259752, A259754.

[2*(6*n+(-1)^n-3): n in [1..60]]; // Vincenzo Librandi, Aug 27 2015

A[n_] := A[n] = Sum[a b, {a, 1,n}, {b, a + 1, n}]; Select[Range[200], Mod[A[#], #]/# == 3/4 &]
Table[2 (6 n + (-1)^n - 3), {n, 1, 60}] (* Bruno Berselli, Oct 23 2015 *)
LinearRecurrence[{1,1,-1},{4,20,28},60] (* Harvey P. Dale, Jul 19 2016 *)

vector(100, n, 2*(6*n+(-1)^n-3)) \\ Altug Alkan, Oct 23 2015

a(n) = 2*(6*n + (-1)^n - 3).
A259748(a(n))/a(n) = 3/4.
a(n) = 4*A007310(n). - Michel Marcus, Sep 22 2015
G.f.: 4*x*(1 + 4*x + x^2) / ((1 + x)*(1 - x)^2). - Bruno Berselli, Oct 23 2015
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/24. - Amiram Eldar, Dec 31 2021
E.g.f.: 2*(2 + (6*x - 3)*exp(x) + exp(-x)). - David Lovler, Sep 05 2022

Better name from Danny Rorabaugh, Oct 22 2015

A259749 Numbers that are congruent to {1,2,5,7,10,11,13,17,19,23} mod 24.

Original entry on oeis.org

1, 2, 5, 7, 10, 11, 13, 17, 19, 23, 25, 26, 29, 31, 34, 35, 37, 41, 43, 47, 49, 50, 53, 55, 58, 59, 61, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 89, 91, 95, 97, 98, 101, 103, 106, 107, 109, 113, 115, 119, 121, 122, 125, 127, 130, 131, 133, 137, 139, 143, 145

Offset: 1

José María Grau Ribas, Jul 04 2015

Original name: Numbers n such that A259748(n) = 0.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-1).

Cf. A000914.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259750, A259751, A259752, A259754, A259755.

A[n_] := A[n] = Sum[a b, {a, 1, n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]  == 0 & ]
Rest@ CoefficientList[Series[x (1 + x^2) (1 + 2 x^2 - x^3 + 2 x^4 - 2 x^5 + 3 x^6 + x^7)/((1 - x)^2*(1 - x + x^2 - x^3 + x^4) (1 + x + x^2 + x^3 + x^4)), {x, 0, 61}], x] (* Michael De Vlieger, Aug 25 2016 *)
Select[Range[150],MemberQ[{1,2,5,7,10,11,13,17,19,23},Mod[#,24]]&] (* or *) LinearRecurrence[{2,-2,2,-2,2,-2,2,-2,2,-1},{1,2,5,7,10,11,13,17,19,23},70] (* Harvey P. Dale, Jan 15 2022 *)

Vec(x*(1+x^2)*(1+2*x^2-x^3+2*x^4-2*x^5+3*x^6+x^7)/((1-x)^2*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)) + O(x^100)) \\ Colin Barker, Aug 25 2016

A259748(a(n)) = Sum_{x*y: x,y in Z/a(n)Z, x<>y} = 0.

G.f.: x*(1+x^2)*(1+2*x^2-x^3+2*x^4-2*x^5+3*x^6+x^7) / ((1-x)^2*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)). - Colin Barker, Aug 25 2016

Better name from Danny Rorabaugh, Oct 22 2015

A259751 Numbers that are congruent to {8, 16} mod 24.

Original entry on oeis.org

8, 16, 32, 40, 56, 64, 80, 88, 104, 112, 128, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 256, 272, 280, 296, 304, 320, 328, 344, 352, 368, 376, 392, 400, 416, 424, 440, 448, 464, 472, 488, 496, 512, 520, 536, 544, 560, 568, 584, 592, 608, 616, 632

Offset: 1

José María Grau Ribas, Jul 06 2015

Original name: Numbers n such that n/A259748(n) = 4.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000914, A001651.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259749, A259750, A259752, A259754, A259755.

A[n_] := A[n] = Sum[a b, {a, 1,  n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]/# == 1/4 & ]

Vec(8*x*(1+x+x^2)/((1-x)^2*(1+x)) + O(x^100)) \\ Colin Barker, Aug 26 2016

A259748(a(n))/a(n) = 1/4.
a(n) = 8*A001651. - Danny Rorabaugh, Oct 22 2015
From Colin Barker, Aug 26 2016: (Start)
a(n) = 12*n-2*(-1)^n-6.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: 8*x*(1+x+x^2) / ((1-x)^2*(1+x)).
(End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/72. - Amiram Eldar, Dec 31 2021
E.g.f.: 2*(4 + (6*x - 3)*exp(x) - exp(-x)). - David Lovler, Sep 05 2022

Better name from Danny Rorabaugh, Oct 22 2015

A259752 a(n) = 24*n - 18.

Original entry on oeis.org

6, 30, 54, 78, 102, 126, 150, 174, 198, 222, 246, 270, 294, 318, 342, 366, 390, 414, 438, 462, 486, 510, 534, 558, 582, 606, 630, 654, 678, 702, 726, 750, 774, 798, 822, 846, 870, 894, 918, 942, 966, 990, 1014, 1038, 1062, 1086, 1110, 1134, 1158, 1182, 1206

Offset: 1

José María Grau Ribas, Jul 12 2015

Original name: Numbers n such that n/A259748(n) = 6.

Partial sums give A152746. - Leo Tavares, Jul 29 2023

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000914, A016813, A063128, A063148, A152746.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259749, A259750, A259751, A259754, A259755.

A[n_] := A[n] = Sum[a b, {a, 1,  n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]/# == 1/6 & ]

A259748(a(n))/a(n) = 1/6.
a(n) = 6*A016813(n-1). - Michel Marcus, Jul 18 2015
G.f.: 6*x*(3*x+1)/(x-1)^2. - Alois P. Heinz, Jul 29 2023
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 6*(exp(x)*(4*x - 3) + 3).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Better name from Danny Rorabaugh, Oct 22 2015

A259754 Numbers that are congruent to {3,9,15,18,21} mod 24.

Original entry on oeis.org

3, 9, 15, 18, 21, 27, 33, 39, 42, 45, 51, 57, 63, 66, 69, 75, 81, 87, 90, 93, 99, 105, 111, 114, 117, 123, 129, 135, 138, 141, 147, 153, 159, 162, 165, 171, 177, 183, 186, 189, 195, 201, 207, 210, 213, 219, 225, 231, 234, 237, 243, 249, 255, 258, 261, 267

Offset: 1

José María Grau Ribas, Jul 12 2015

Original name: Numbers n such that n/A259748(n) = 3/2.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A000914.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259749, A259750, A259751, A259752, A259755.

A[n_] := A[n] = Sum[a b, {a, 1, n}, {b, a + 1, n}]; Select[Range[200], Mod[A[#], #]/# == 2/3 &]
Rest@ CoefficientList[Series[3 x (1 + x) (1 + x + x^2 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)), {x, 0, 56}], x] (* Michael De Vlieger, Aug 25 2016 *)
LinearRecurrence[{1,0,0,0,1,-1},{3,9,15,18,21,27},60] (* Harvey P. Dale, Aug 30 2016 *)

Vec(3*x*(1+x)*(1+x+x^2+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)) + O(x^100)) \\ Colin Barker, Aug 25 2016

A259748(a(n))/a(n) = 2/3.
a(n) = 3*A047584(n). - Michel Marcus, Jul 18 2015
From Colin Barker, Aug 25 2016: (Start)
a(n) = a(n-1)+a(n-5)-a(n-6) for n>6.
G.f.: 3*x*(1+x)*(1+x+x^2+x^4) / ((1-x)^2*(1+x+x^2+x^3+x^4)).
(End)

Better name from Danny Rorabaugh, Oct 22 2015

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A008606 Multiples of 24.

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A073762 a(n) = 24*n - 12.

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A259750 Numbers that are congruent to {14, 22} mod 24.

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A259755 Numbers that are congruent to {4, 20} mod 24.

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A259749 Numbers that are congruent to {1,2,5,7,10,11,13,17,19,23} mod 24.

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A259751 Numbers that are congruent to {8, 16} mod 24.

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A259752 a(n) = 24*n - 18.

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