A008588 -id:A008588 - cp's OEIS frontend

A008615 a(n) = floor(n/2) - floor(n/3).

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14

Offset: 0

N. J. A. Sloane, Simon Plouffe

If the two leading 0's are dropped, this becomes the essentially identical sequence A103221, with g.f. 1/((1-x^2)*(1-x^3)), which arises in many contexts. For example, 1/((1-x^4)*(1-x^6)) is the Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+8 cusp forms for Gamma_0( 1 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of ways n can be written as the sum of a positive even number and a nonnegative multiple of 3 and so the number of ways (n-2) can be written as the sum of a nonnegative even number and a nonnegative multiple of 3 and also the number of ways (n+3) can be written as the sum of a positive even number and a positive multiple of 3.
It appears that this is also the number of partitions of 2n+6 that are 4-term arithmetic progressions. - John W. Layman, May 01 2009 [verified by Wesley Ivan Hurt, Jan 17 2021]
a(n) is the number of (n+3)-digit fixed points under the base-3 Kaprekar map A164993 (see A164997 for the list of fixed points). - Joseph Myers, Sep 04 2009
Starting from n=10 also the number of balls in new consecutive hexagonal edges, if an (infinite) chain of balls is winded spirally around the first ball at the center, such that each six steps make an entire winding. - K. G. Stier, Dec 21 2012
In any three consecutive terms at least two of them are equal to each other. - Michael Somos, Mar 01 2014
Number of partitions of (n-2) into parts 2 and 3. - David Neil McGrath, Sep 05 2014
a(n), n >= 0, is also the dimension of S_{2*(n+4)}, the complex vector space of modular cusp forms of weight 2*(n+4) and level 1 (full modular group). The dimension of S_0, S_2, S_4 and S_6 is 0. See, e.g., Ash and Gross, p. 178. Table 13.1. - Wolfdieter Lang, Sep 16 2016
From Wolfdieter Lang, May 08 2017: (Start)
a(n-2) = floor((n-2)/2) - floor((n-2)/3) = floor(n/2) - floor((n+1)/3) is for n >=0 the number of integers k in the interval (n+1)/3 < k <= floor(n/2). This problem appears in the computation of the number of zeros of Chebyshev S(n, x) polynomials (coefficients in A049310) in the open interval (-1, +1). See a comment there. This computation was motivated by a conjecture given in A008611 by Michel Lagneau, Mar 31 2017.
a(n) is also the number of integers k in the closed interval (n+1)/3 <= k <= floor(n/2), which is floor(n/2) - (ceiling((n+1)/3) - 1) for n >= 0 (proof trivial for n+1 == 0 (mod 3) and otherwise). From the preceding statement this a(n) is also a(n-2) + [n == 2 (mod 3)] for n >= 0 (with [statement] = 1 if the statement is true and zero otherwise). This proves the recurrence given by Michael Somos in the formula section. (End)
Assuming the Collatz conjecture to be true, for n > 1, a(n+7) is the row length of the n-th row of A340985. That is, the number of weakly connected components of the Collatz digraph of order n. - Sebastian Karlsson, Feb 23 2021

			G.f. = x^2 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + ...

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
J.-M. Kantor, Où en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
David Broadhurst, Feynman integrals, L-series and Kloosterman moments, arXiv:1604.03057 [physics.gen-ph], 2016. See Cor. 1.
J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 212
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 448
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
William A. Stein, The modular forms database
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
James Tanton, Young students approach integer triangles, FOCUS 22 no. 5 (2002), 4 - 6.
James Tanton et al., Young students approach integer triangles, FOCUS 22 no. 5 (2002), 4 - 6.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Essentially the same as A103221.
First differences of A069905 (and A001399).
Cf. A002264, A004009, A004526, A005044, A008588, A008611, A013973, A016921, A016933, A016957, A049310, A057078, A164993, A164997, A340985.

a008615 n = n `div` 2 - n `div` 3  -- Reinhard Zumkeller, Apr 28 2014

[Floor(n/2)-Floor(n/3): n in [0..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

a := func< n | n lt 2 select 0 else n eq 2 select 1 else Dimension( ModularForms( PSL2( Integers()), 2*n-4))>; /* Michael Somos, Dec 11 2018 */

a := n-> floor(n/2) - floor(n/3): seq(a(n), n = 0 .. 87);

a[n_]:=Floor[n/2]-Floor[n/3]; Array[a,90,0] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008; corrected by Harvey P. Dale, Nov 30 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1}, {0, 0, 1, 0, 1}, 100]; (* Vincenzo Librandi, Sep 09 2015 *)

{a(n) = (n\2) - (n\3)}; /* Michael Somos, Feb 06 2003 */

def A008615(n): return n//2 - n//3 # Chai Wah Wu, Jun 07 2022

a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: x^2 / ((1-x^2) * (1-x^3)).
From Reinhard Zumkeller, Feb 27 2008: (Start)
a(A016933(n)) = a(A016957(n)) = a(A016969(n)) = n+1.
a(A008588(n)) = a(A016921(n)) = a(A016945(n)) = n. (End)
a(6*k) = k, k >= 0. - Zak Seidov, Sep 09 2012
a(n) = A005044(n+1) - A005044(n-3). - Johannes W. Meijer, Oct 18 2013
a(n) = floor((n+4)/6) - floor((n+3)/6) + floor((n+2)/6). - Mircea Merca, Nov 27 2013
Euler transform of length 3 sequence [0, 1, 1]. - Michael Somos, Mar 01 2014
a(n+2) = a(n) + 1 if n == 0 (mod 3), a(n+2) = a(n) otherwise. - Michael Somos, Mar 01 2014. See the May 08 2017 comment above. - Wolfdieter Lang, May 08 2017
a(n) = -a(-1 - n) for all n in Z. - Michael Somos, Mar 01 2014.
a(n) = A004526(n) - A002264(n). - Reinhard Zumkeller, Apr 28 2014
a(n) = Sum_{i=0..n-2} (floor(i/6)-floor((i-3)/6))*(-1)^i. - Wesley Ivan Hurt, Sep 08 2015
a(n) = a(n+6) - 1 = A103221(n+4) - 1, n >= 0. - Wolfdieter Lang, Sep 16 2016
12*a(n) = 2*n +1 +3*(-1)^n -4*A057078(n). - R. J. Mathar, Jun 19 2019
a(n) = Sum_{k=1..floor((n+3)/2)} Sum_{j=k..floor((2*n+6-k)/3)} Sum_{i=j..floor((2*n+6-j-k)/2)} ([j-k = i-j = 2*n+6-2*i-j-k] - [k = j = i = 2*n+6-i-j-k]), where [ ] is the (generalized) Iverson bracket. - Wesley Ivan Hurt, Jan 17 2021
E.g.f.: (3*(2 + x)*cosh(x) - 2*exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*(x-1)*sinh(x))/18. - Stefano Spezia, Oct 17 2022

A016933 a(n) = 6*n + 2.

Original entry on oeis.org

2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326

Offset: 0

N. J. A. Sloane

Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
Exponents n>1 for which 1 - x + x^n is reducible. - Ron Knott, Oct 13 2016
For the Collatz problem, these are the descenders' values that require division by 2. - Fred Daniel Kline, Jan 19 2017
For n > 3, also the number of (not necessarily maximal) cliques in the n-helm graph. - Eric W. Weisstein, Nov 29 2017

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See p. 9.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Helm Graph.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008588, A016921, A016945, A016957, A016969, A017569, A016777, A016813.

a016933 = (+ 2) . (* 6)  -- Reinhard Zumkeller, Jul 05 2013

a[1]:=2:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..47); # Zerinvary Lajos, Mar 16 2008

Range[2, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[6 n + 2, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
6 Range[0, 20] + 2 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{2, -1}, {8, 14}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[2 (1 + 2 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=6*n+2 \\ Charles R Greathouse IV, Jul 10 2016

[i+2 for i in range(280) if gcd(i,6) == 6] # Zerinvary Lajos, May 20 2009

A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A013730(n). - Reinhard Zumkeller, Feb 24 2009
A089911(2*a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013
a(n) = 2*(6*n-1) - a(n-1) (with a(0)=2). - Vincenzo Librandi, Nov 20 2010
G.f.: 2*(1+2*x)/(1-x)^2. - Colin Barker, Jan 08 2012
a(n) = (3 * A016813(n) + 1) / 2.- Fred Daniel Kline, Jan 20 2017
a(n) = A016789(A005843(n)). - Felix Fröhlich, Jan 20 2017
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 + log(2)/6. - Amiram Eldar, Dec 10 2021
a(n) = 2 * A016777(n). - Alois P. Heinz, Dec 27 2023
From Elmo R. Oliveira, Mar 08 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
E.g.f.: 2*exp(x)*(1 + 3*x). (End)

A122841 Greatest k such that 6^k divides n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Sep 13 2006

nonn

See A054895 for the partial sums. - Hieronymus Fischer, Jun 08 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A122840, A007814, A007949, A054895, A234959.

a122841 = f 0 where
   f y x = if r > 0 then y else f (y + 1) x'
           where (x', r) = divMod x 6
-- Reinhard Zumkeller, Nov 10 2013

Table[IntegerExponent[n, 6], {n, 1, 100}] (* Amiram Eldar, Sep 14 2020 *)

a(n) = valuation(n, 6); \\ Michel Marcus, Jan 17 2022

From Hieronymus Fischer, Jun 03 2012: (Start)
With m = floor(log_6(n)), frac(x) = x-floor(x):
a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/6^j))).
a(n) = m + Sum_{j=1..m} (floor(-frac(n/6^j))).
a(n) = A054895(n) - A054895(n-1).
G.f.: Sum_{j>0} x^6^j/(1-x^6^j). (End)
a(A047253(n)) = 0; a(A008588(n)) > 0; a(A044102(n)) > 1. - Reinhard Zumkeller, Nov 10 2013
6^a(n) = A234959(n), n >= 1. - Wolfdieter Lang, Jun 30 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/5. - Amiram Eldar, Jan 17 2022
a(n) = min(A007814(n), A007949(n)). - Jianing Song, Jul 23 2022

A028896 6 times triangular numbers: a(n) = 3n(n+1).

Original entry on oeis.org

0, 6, 18, 36, 60, 90, 126, 168, 216, 270, 330, 396, 468, 546, 630, 720, 816, 918, 1026, 1140, 1260, 1386, 1518, 1656, 1800, 1950, 2106, 2268, 2436, 2610, 2790, 2976, 3168, 3366, 3570, 3780, 3996, 4218, 4446, 4680, 4920, 5166, 5418, 5676

Offset: 0

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0, 6, ...
The spiral begins:
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
    <==90==60==36==18===6===0   3  12  27  48  75
           /   /   /   /   /   /   /   /   /   /
         61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
(End)
If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007
a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008
Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - Omar E. Pol, Sep 18 2011
Partial sums of A008588. - R. J. Mathar, Aug 28 2014
Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
a(n-4) is the maximum irregularity over all maximal 3-degenerate graphs with n vertices.  The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices).  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Leo Tavares, Illustration: Centroid Hexagons.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000567, A003215, A008588, A024966, A028895, A033996, A046092, A049598, A084939, A084940, A084941, A084942, A084943, A084944, A124080.
Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A152773 (6-cycles).
Cf. A007531.
The partial sums give A007531. - Leo Tavares, Jan 22 2022
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

List([0..44],n->3*n*(n+1)); # Muniru A Asiru, Mar 15 2019

[3*n*(n+1): n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

[seq(6*binomial(n,2),n=1..44)]; # Zerinvary Lajos, Nov 24 2006

6 Accumulate[Range[0, 50]] (* Harvey P. Dale, Mar 05 2012 *)
6 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
LinearRecurrence[{3, -3, 1}, {0, 6, 18}, 20] (* Eric W. Weisstein, Jul 27 2017 *)

a(n)=3*n*(n+1) \\ Charles R Greathouse IV, Sep 24 2015

first(n) = Vec(6*x/(1 - x)^3 + O(x^n), -n) \\ Iain Fox, Feb 14 2018

def A028896(n): return 3*n*(n+1) # Chai Wah Wu, Aug 07 2025

O.g.f.: 6*x/(1 - x)^3.
E.g.f.: 3*x*(x + 2)*exp(x). - G. C. Greubel, Aug 19 2017
a(n) = 6*A000217(n).
a(n) = polygorial(3, n+1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003
From Zerinvary Lajos, Mar 06 2007: (Start)
a(n) = A049598(n)/2.
a(n) = A124080(n) - A046092(n).
a(n) = A033996(n) - A002378(n). (End)
a(n) = A002378(n)*3 = A045943(n)*2. - Omar E. Pol, Dec 12 2008
a(n) = a(n-1) + 6*n for n>0, a(0)=0. - Vincenzo Librandi, Aug 05 2010
a(n) = A003215(n) - 1. - Omar E. Pol, Oct 03 2011
From Philippe Deléham, Mar 26 2013: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=6, a(2)=18.
a(n) = A174709(6*n + 5). (End)
a(n) = A049450(n) + 4*n. - Lear Young, Apr 24 2014
a(n) = Sum_{i = n..2*n} 2*i. - Bruno Berselli, Feb 14 2018
a(n) = A320047(1, n, 1). - Kolosov Petro, Oct 04 2018
a(n) = T(3*n) - T(2*n-2) + T(n-2), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 04 2020
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 1/3. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(3/Pi)*cos(sqrt(7/3)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (3/Pi)*cosh(Pi/(2*sqrt(3))). (End)

A224613 a(n) = sigma(6*n).

Original entry on oeis.org

12, 28, 39, 60, 72, 91, 96, 124, 120, 168, 144, 195, 168, 224, 234, 252, 216, 280, 240, 360, 312, 336, 288, 403, 372, 392, 363, 480, 360, 546, 384, 508, 468, 504, 576, 600, 456, 560, 546, 744, 504, 728, 528, 720, 720, 672, 576, 819, 684, 868, 702, 840, 648

Offset: 1

Zak Seidov, Apr 22 2013

nonn

Conjectures: sigma(6n) > sigma(6n - 1) and sigma(6n) > sigma(6n + 1).
Conjectures are false. Try prime 73961483429 for n. One finds sigma(6*73961483429) < sigma(6*73961483429+1). The number n = 105851369791 provides a counterexample for the other case. - T. D. Noe, Apr 22 2013
Sum of the divisors of the numbers k which have the property that the width associated to the vertex of the first (also the last) valley of the smallest Dyck path of the symmetric representation of sigma(k) is equal to 2 (see example). Other positive integers have width 0 or 1 associated to the mentioned valley. - Omar E. Pol, Aug 11 2021

			From _Omar E. Pol_, Aug 11 2021: (Start)
Illustration of initial terms:
----------------------------------------------------------------------
   n    6*n   a(n)    Diagram:  1           2           3           4
----------------------------------------------------------------------
                                _           _           _           _
                               | |         | |         | |         | |
                               | |         | |         | |         | |
                          * _ _| |         | |         | |         | |
                           |  _ _|         | |         | |         | |
                      _ _ _| |_|           | |         | |         | |
   1     6     12    |_ _ _ _|      * _ _ _| |         | |         | |
                                    _|  _ _ _|         | |         | |
                                * _|  _| |             | |         | |
                                 |  _|  _|    * _ _ _ _| |         | |
                                 | |_ _|       |  _ _ _ _|         | |
                      _ _ _ _ _ _| |          _| | |               | |
   2    12     28    |_ _ _ _ _ _ _|        _|  _|_|    * _ _ _ _ _| |
                                      * _ _|  _|         |  _ _ _ _ _|
                                       |  _ _|        _ _| | |
                                       | |_ _|      _|  _ _| |
                                       | |        _|  _|  _ _|
                      _ _ _ _ _ _ _ _ _| |       |  _|  _|
   3    18     39    |_ _ _ _ _ _ _ _ _ _|  * _ _| |  _|
                                             |  _ _| |
                                             | |_ _ _|
                                             | |
                                             | |
                      _ _ _ _ _ _ _ _ _ _ _ _| |
   4    24     60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the mentioned vertices are aligned on two straight lines that meet at point (3,3).
a(n) equals the area (also the number of cells) in the n-th diagram. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), this sequence (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).
Cf. A000203 (sigma(n)), A053224 (n: sigma(n) < sigma(n+1)).
Cf. A067825 (even n: sigma(n)< sigma(n+1)).
Cf. A008588, A237593.

DivisorSigma[1,6*Range[60]] (* Harvey P. Dale, Apr 16 2016 *)

a(n)=sigma(6*n) \\ Charles R Greathouse IV, Apr 22 2013

from sympy import divisor_sigma
def a(n):  return divisor_sigma(6*n)
print([a(n) for n in range(1, 54)]) # Michael S. Branicky, Dec 28 2021

from math import prod
from collections import Counter
from sympy import factorint
def A224613(n): return prod((p**(e+1)-1)//(p-1) for p, e in (Counter(factorint(n))+Counter([2,3])).items()) # Chai Wah Wu, Sep 07 2023

a(n) = A000203(6n).
a(n) = A000203(A008588(n)). - Omar E. Pol, Aug 11 2021
Sum_{k=1..n} a(k) = (55*Pi^2/72) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

Corrected by Harvey P. Dale, Apr 16 2016

A121026 Multiples of 6 containing a 6 in their decimal representation.

Original entry on oeis.org

6, 36, 60, 66, 96, 126, 156, 162, 168, 186, 216, 246, 264, 276, 306, 336, 360, 366, 396, 426, 456, 462, 468, 486, 516, 546, 564, 576, 600, 606, 612, 618, 624, 630, 636, 642, 648, 654, 660, 666, 672, 678, 684, 690, 696, 726, 756, 762, 768, 786, 816, 846, 864

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Index entries for 10-automatic sequences.

Intersection of A008588 and A011536.

Cf. A121041, A011531, A121022, A121023, A121024, A121025, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Select[6*Range[200],DigitCount[#,10,6]>0&] (* Harvey P. Dale, Jan 07 2021 *)

is(n)=n%6==0 && setsearch(Set(digits(n)), 6) \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 6n. - Charles R Greathouse IV, Feb 12 2017

Corrected by T. D. Noe, Oct 25 2006

Typo in comment fixed by Reinhard Zumkeller, May 01 2011

A165998 Denominators of Taylor series expansion of 1/(3x)log((1+x)/(1-x)^2).

Original entry on oeis.org

1, 6, 3, 12, 5, 18, 7, 24, 9, 30, 11, 36, 13, 42, 15, 48, 17, 54, 19, 60, 21, 66, 23, 72, 25, 78, 27, 84, 29, 90, 31, 96, 33, 102, 35, 108, 37, 114, 39, 120, 41, 126, 43, 132, 45, 138, 47, 144, 49, 150, 51, 156, 53, 162, 55, 168, 57, 174, 59, 180, 61, 186, 63, 192, 65, 198

Offset: 0

Jaume Oliver Lafont, Oct 03 2009

Numerators are all 1.
Setting x=1/3 into 1/(3*x)*log((1+x)/(1-x)^2) = Sum_{k>=0} x^k/((2-(-1)^k)*(k+1)),
log(3) = Sum_{k>=0} 1/((2-(-1)^k)*(k+1)*3^k) = Sum_{k>=0} (9/(2k+1)+1/(2k+2))/9^(k+1) is obtained.
It appears that this is also the first differences of the generalized decagonal numbers A074377. - Omar E. Pol, Sep 10 2011
It appears that this is also A005408 and positive terms of A008588 interleaved. - Omar E. Pol, May 28 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1004
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A005408, A008588, A074377, A154920.

[(2-(-1)^n)*(n+1): n in [0..350]]; // Vincenzo Librandi, Apr 04 2011

LinearRecurrence[{0,2,0,-1}, {1, 6, 3, 12}, 50] (* Vincenzo Librandi, Feb 22 2012 *)

PARI
```
a(n)=(2-(-1)^n)*(n+1)
```

G.f.: (1+6*x+x^2)/(1-x^2)^2.
a(n) = (2-(-1)^n)*(n+1) (see PARI's code by Jaume Oliver Lafont).
a(2n)= 2n+1. a(2n+1) = 6*(n+1). - R. J. Mathar, Apr 02 2011
With offset 1 this sequence is multiplicative (in fact, a generalized totient function): a(p^e) = p^e for any odd prime p and a(2^e) = 3*2^e for e >= 1. - Charles R Greathouse IV, Mar 09 2015
With offset 1, Dirichlet g.f.: zeta(s-1) * (1 + 2^(2-s)). - Amiram Eldar, Oct 25 2023

A249674 a(n) = 30*n.

Original entry on oeis.org

0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1260, 1290, 1320, 1350, 1380, 1410, 1440

Offset: 0

Kaylan Purisima, Nov 03 2014

Numbers divisible by 2, 3 and 5. - Robert Israel, Nov 19 2014

a(n) is the maximum score of a 10-pin n-frame bowling game and the maximum score of an n-pin 10-frame bowling game, given the rules: a strike is worth the number of pins in each frame plus the number of pins knocked down by the next two balls (except in the last frame), a spare is worth the number of pins in each frame plus the number of pins knocked down by the next ball (except in the last frame), and if a strike or spare is earned in the last frame then the player must continue to throw balls until they have thrown 3 balls in the last frame. - Iain Fox, Mar 02 2018

			a(7) = 7 * 30 = 210.

Iain Fox, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Bowling.
Wikipedia, Ten-pin bowling.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A008585, A008587, A008588, A008592, A008597, A050519, A169823.

[30*n : n in [0..50]]; // Wesley Ivan Hurt, Nov 18 2014

A249674:=n->30*n: seq(A249674(n), n=0..50); # Wesley Ivan Hurt, Nov 18 2014

30*Range[0, 59] (* Alonso del Arte, Nov 18 2014 *)

vector(100,n,30*(n-1)) \\ Derek Orr, Nov 18 2014

first(n) = Vec(30*x/(x-1)^2 + O(x^n), -n) \\ Iain Fox, Mar 02 2018

G.f.: 30*x/(x-1)^2; a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Nov 18 2014
a(n) = 2*A008597(n) = 3*A008592(n) = 5*A008588(n) = 6*A008587(n) = 10*A008585(n) = 15*A005843(n). - Omar E. Pol, Nov 24 2014
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 30*x*exp(x).
a(n) = A169823(n)/2. (End)

A010722 Constant sequence: the all 6's sequence.

Original entry on oeis.org

6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane

Continued fraction expansion of 3+sqrt(10). - Bruno Berselli, Mar 15 2011
Decimal expansion of Sum_{n >= 0} n/binomial(2*n+1, n) = 2/3. - Bruno Berselli, Sep 14 2015
Decimal expansion of 2/3. - Franklin T. Adams-Watters, Feb 23 2019

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1014.
Tanya Khovanova, Recursive Sequences.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A145429: decimal expansion of Sum_{n >= 0} n/binomial(2*n, n).
Cf. A000012, A007395, A010701, A010709, A010716.
First differences of A008588.

a(n)=6 \\ Charles R Greathouse IV, Sep 28 2015

G.f.: 6/(1-x). - Bruno Berselli, Mar 15 2011
E.g.f.: 6*e^x. - Vincenzo Librandi, Jan 27 2012
a(n) = floor(1/(-n + csc(1/n))). - Clark Kimberling, Mar 10 2020

A097325 Period 6: repeat [0, 1, 1, 1, 1, 1].

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 0

Ralf Stephan, Aug 16 2004

a(n) is 0 if 6 divides n, 1 otherwise.

Antti Karttunen, Table of n, a(n) for n = 0..26244
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Characteristic sequence of A047253.
Binary complement of A079979.
Cf. A010875, A168185, A145568, A168184, A168182, A168181, A109720, A011558, A166486, A011655, A000035.

[Sign(n mod 6) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

seq(signum(k mod 6), k=0..100); # Wesley Ivan Hurt, Jun 29 2013

Table[Boole[Not[Divisible[n, 6]]], {n, 0, 89}] (* Alonso del Arte, Oct 21 2013 *)
PadRight[{}, 120, {0, 1, 1, 1, 1, 1}] (* Michael De Vlieger, Dec 22 2017 *)

PARI
```
a(n) = sign(n%6);
```

(define (A097325 n) (if (zero? (modulo n 6)) 0 1)) ;; Antti Karttunen, Dec 22 2017

G.f.: 1/(1-x) - 1/(1-x^6) = Sum_{k>=0} x^k - x^(6*k).
Recurrence: a(n+6) = a(n), a(0) = 0, a(i) = 1, 1 <= i <= 5.
a(n) = (1/4) * (3 - (-1)^n - (-1)^((n+1)/3) - (-1)^((2n+1)/3)).
From Reinhard Zumkeller, Nov 30 2009: (Start)
a(n) = 1 - A079979(n).
a(A047253(n)) = 1, a(A008588(n)) = 0.
A033438(n) = Sum_{k=0..n} a(k)*(n-k). (End)
Dirichlet g.f.: (1 - 1/6^s)*zeta(s). - R. J. Mathar, Feb 19 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m, n > 0. - Boris Putievskiy, May 08 2013
a(n) = sign(n mod 6). - Wesley Ivan Hurt, Jun 29 2013
a(n) = ceiling(5n/6) - floor(5n/6). - Wesley Ivan Hurt, Jun 20 2014

New name from Omar E. Pol, Oct 21 2013

cp's OEIS Frontend

A008615 a(n) = floor(n/2) - floor(n/3).

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A016933 a(n) = 6*n + 2.

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A122841 Greatest k such that 6^k divides n.

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A028896 6 times triangular numbers: a(n) = 3*n*(n+1).

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A224613 a(n) = sigma(6*n).

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A121026 Multiples of 6 containing a 6 in their decimal representation.

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A028896 6 times triangular numbers: a(n) = 3n(n+1).

A165998 Denominators of Taylor series expansion of 1/(3x)log((1+x)/(1-x)^2).