A010766 -id:A010766 - cp's OEIS frontend

A008619 Positive integers repeated.

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38

Offset: 0

N. J. A. Sloane

The floor of the arithmetic mean of the first n+1 positive integers. - Cino Hilliard, Sep 06 2003
Number of partitions of n into powers of 2 where no power is used more than three times, or 4th binary partition function (see A072170).
Number of partitions of n in which the greatest part is at most 2. - Robert G. Wilson v, Jan 11 2002
Number of partitions of n into at most 2 parts. - Jon Perry, Jun 16 2003
a(n) = #{k=0..n: k+n is even}. - Paul Barry, Sep 13 2003
Number of symmetric Dyck paths of semilength n+2 and having two peaks. E.g., a(6)=4 because we have UUUUUUU*DU*DDDDDDD, UUUUUU*DDUU*DDDDDD, UUUUU*DDDUUU*DDDDD and UUUU*DDDDUUUU*DDDD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004
Smallest positive integer whose harmonic mean with another positive integer is n (for n > 0). For example, a(6)=4 is already given (as 4 is the smallest positive integer such that the harmonic mean of 4 (with 12) is 6) - but the harmonic mean of 2 (with -6) is also 6 and 2 < 4, so the two positive integer restrictions need to be imposed to rule out both 2 and -6.
Second outermost diagonal of Losanitsch's triangle (A034851). - Alonso del Arte, Mar 12 2006
Arithmetic mean of n-th row of A080511. - Amarnath Murthy, Mar 20 2003
a(n) is the number of ways to pay n euros (or dollars) with coins of one and two euros (respectively dollars). - Richard Choulet and Robert G. Wilson v, Dec 31 2007
Inverse binomial transform of A045623. - Philippe Deléham, Dec 30 2008
Coefficient of q^n in the expansion of (m choose 2)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
This Itakura comment follows from a partial fraction decomposition (m choose 2)q = [(1-q^(2m-2))/(1+q) + (1-q^(2m-2))/(1-q) +2 (1-q^(m-1))^2/(1-q)^2]/4. Interpreted as generating functions in q, they have convolution structures; the first term in the numerator creates +1,-1,+1,-1 etc, the 2nd term creates +1,+1,+1,+1 etc., the 3rd term 2,4,6,8, etc. as m->infinity. - _R. J. Mathar, Sep 25 2008
Binomial transform of (-1)^n*A034008(n) = [1,0,1,-2,4,-8,16,-32,...]. - Philippe Deléham, Nov 15 2009
From Jon Perry_, Nov 16 2010: (Start)
Column sums of:
  1 1 1 1 1 1...
      1 1 1 1...
          1 1...
  ..............
  --------------
  1 1 2 2 3 3...  (End)
This sequence is also the half-convolution of the powers of 1 sequence A000012 with itself. For the definition of half-convolution see a comment on A201204, where also the rule for the o.g.f. is given. - Wolfdieter Lang, Jan 09 2012
a(n) is also the number of roots of the n-th Bernoulli polynomial in the right half-plane for n>0. - Michel Lagneau, Nov 08 2012
a(n) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the Exe vibronic perturbation matrix, H(Q) (cf. Viel & Eisfeld). - Bradley Klee, Jul 21 2015
a(n) is the number of distinct integers in the n-th row of Pascal's triangle. - Melvin Peralta, Feb 03 2016
a(n+1) for n >= 3 is the diameter of the Generalized Petersen Graph G(n, 1). - Nick Mayers, Jun 06 2016
The arithmetic function v_1(n,2) as defined in A289198. - Robert Price, Aug 22 2017
Also, this sequence is the second column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - Mohammad K. Azarian, Jul 18 2018
a(n+2) is the least k such that given any k integers, there exist two of them whose sum or difference is divisible by n. - Pablo Hueso Merino, May 09 2020
Column k = 2 of A051159. - John Keith, Jun 28 2021

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 109, Eq. [6c]; p. 116, P(n,2).
D. Parisse, 'The tower of Hanoi and the Stern-Brocot Array', Thesis, Munich 1997

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 120
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 209
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 351
Gerzson Keri and Patric R. J. Östergård, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
L. F. Klosinski, G. L. Alexanderson and A. P. Hillman, The William Lowell Putnam Mathematical Competition, Amer. Math. Monthly 91 (1984), 487-495. See Problem B2.
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, arXiv:2309.04012 [math.NT], 2023.
Bruce Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.
Alexandra Viel and Wolfgang Eisfeld, Effects of higher order Jahn-Teller coupling on the nuclear dynamics, J. Chem. Phys., 120, 4603 (2004).
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature
Index entries for sequences related to Stern's sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for Molien series

Essentially same as A004526.
Harmonic mean of a(n) and A056136 is n.
Cf. A001057, A065033, A001399, A001400, A001401.
a(n)=A010766(n+2, 2).
Cf. A010551 (partial products).
Cf. A263997 (a block spiral).
Cf. A289187.
Column 2 of A235791.

a008619 = (+ 1) . (`div` 2)
a008619_list = concatMap (\x -> [x,x]) [1..]
-- Reinhard Zumkeller, Apr 02 2012

I:=[1,1,2]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..100]]; // Vincenzo Librandi, Feb 04 2015

a:= n-> iquo(n+2, 2): seq(a(n), n=0..75);

Flatten[Table[{n,n},{n,35}]] (* Harvey P. Dale, Sep 20 2011 *)
With[{c=Range[40]},Riffle[c,c]] (* Harvey P. Dale, Feb 23 2013 *)
CoefficientList[Series[1/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *)
LinearRecurrence[{1, 1, -1}, {1, 1, 2}, 75] (* Robert G. Wilson v, Feb 05 2015 *)
Table[QBinomial[n, 2, -1], {n, 2, 75}] (* John Keith, Jun 28 2021 *)

PARI
```
a(n)=n\2+1
```

def A008619(n): return (n>>1)+1 # Chai Wah Wu, Jul 07 2022

a = lambda n: 1 if n==0 else a(n-1)+1 if 2.divides(n) else a(n-1) # Peter Luschny, Feb 05 2015

(2 to 99).map( / 2) // _Alonso del Arte, May 09 2020

Euler transform of [1, 1].
a(n) = 1 + floor(n/2).
G.f.: 1/((1-x)(1-x^2)).
E.g.f.: ((3+2*x)*exp(x) + exp(-x))/4.
a(n) = a(n-1) + a(n-2) - a(n-3) = -a(-3-n).
a(0) = a(1) = 1 and a(n) = floor( (a(n-1) + a(n-2))/2 + 1 ).
a(n) = (2*n + 3 + (-1)^n)/4. - Paul Barry, May 27 2003
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} binomial(j, i)*(-2)^i. - Paul Barry, Aug 26 2003
E.g.f.: ((1+x)*exp(x) + cosh(x))/2. - Paul Barry, Sep 13 2003
a(n) = A108299(n-1,n)*(-1)^floor(n/2) for n > 0. - Reinhard Zumkeller, Jun 01 2005
a(n) = A108561(n+2,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
a(n) = A125291(A125293(n)) for n>0. - Reinhard Zumkeller, Nov 26 2006
a(n) = ceiling(n/2), n >= 1. - Mohammad K. Azarian, May 22 2007
INVERT transformation yields A006054 without leading zeros. INVERTi transformation yields negative of A124745 with the first 5 terms there dropped. - R. J. Mathar, Sep 11 2008
a(n) = A026820(n,2) for n > 1. - Reinhard Zumkeller, Jan 21 2010
a(n) = n - a(n-1) + 1 (with a(0)=1). - Vincenzo Librandi, Nov 19 2010
a(n) = A000217(n) / A110654(n). - Reinhard Zumkeller, Aug 24 2011
a(n+1) = A181971(n,n). - Reinhard Zumkeller, Jul 09 2012
1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction))))) = 1/(e-1), see A073333. - Philippe Deléham, Mar 09 2013
a(n) = floor(A000217(n)/n), n > 0. - L. Edson Jeffery, Jul 26 2013
a(n) = n*a(n-1) mod (n+1) = -a(n-1) mod (n+1), the least positive residue modulo n+1 for each expression for n > 0, with a(0) = 1 (basically restatements of Vincenzo Librandi's formula). - Rick L. Shepherd, Apr 02 2014
a(n) = (a(0) + a(1) + ... + a(n-1))/a(n-1), where a(0) = 1. - Melvin Peralta, Jun 16 2015
a(n) = Sum_{k=0..n} (-1)^(n-k) * (k+1). - Rick L. Shepherd, Sep 18 2020
a(n) = a(n-2) + 1 for n >= 2. - Vladimír Modrák, Sep 29 2020
a(n) = A004526(n)+1. - Chai Wah Wu, Jul 07 2022

Additional remarks from Daniele Parisse
Edited by N. J. A. Sloane, Sep 06 2009
Partially edited by Joerg Arndt, Mar 11 2010

A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

Original entry on oeis.org

1, 4, 8, 15, 21, 33, 41, 56, 69, 87, 99, 127, 141, 165, 189, 220, 238, 277, 297, 339, 371, 407, 431, 491, 522, 564, 604, 660, 690, 762, 794, 857, 905, 959, 1007, 1098, 1136, 1196, 1252, 1342, 1384, 1480, 1524, 1608, 1686, 1758, 1806, 1930, 1987, 2080, 2152

Offset: 1

Clark Kimberling

Row sums of triangle A128489. E.g., a(5) = 15 = (10 + 3 + 1 + 1), sum of row 4 terms of triangle A128489. - Gary W. Adamson, Jun 03 2007
Row sums of triangle A134867. - Gary W. Adamson, Nov 14 2007
a(10^4) = 82256014, a(10^5) = 8224740835, a(10^6) = 822468118437, a(10^7) = 82246711794796; see A072692. - M. F. Hasler, Nov 22 2007
Equals row sums of triangle A158905. - Gary W. Adamson, Mar 29 2009
n is prime if and only if a(n) - a(n-1) - 1 = n. - Omar E. Pol, Dec 31 2012
Also the alternating row sums of A236104. - Omar E. Pol, Jul 21 2014
a(n) is also the total number of parts in all partitions of the positive integers <= n into equal parts. - Omar E. Pol, Apr 30 2017
a(n) is also the total area of the terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Nov 04 2017
a(n) is also the area under the Dyck path described in the n-th row of A237593 (see example). - Omar E. Pol, Sep 17 2018
From Omar E. Pol, Feb 17 2020: (Start)
Convolution of A340793 and A000027.
Convolved with A340793 gives A000385. (End)
a(n) is also the number of cubic cells (or cubes) in the n-th level starting from the top of the stepped pyramid described in A245092. - Omar E. Pol, Jan 12 2022

			From _Omar E. Pol_, Aug 20 2021: (Start)
For n = 6 the sum of all divisors of the first six positive integers is [1] + [1 + 2] + [1 + 3] + [1 + 2 + 4] + [1 + 5] + [1 + 2 + 3 + 6] = 1 + 3 + 4 + 7 + 6 + 12 = 33, so a(6) = 33.
On the other hand the area under the Dyck path of the 6th diagram as shown below is equal to 33, so a(6) = 33.
Illustration of initial terms:                        _ _ _ _
                                        _ _ _        |       |_
                            _ _ _      |     |       |         |_
                  _ _      |     |_    |     |_ _    |           |
          _ _    |   |_    |       |   |         |   |           |
    _    |   |   |     |   |       |   |         |   |           |
   |_|   |_ _|   |_ _ _|   |_ _ _ _|   |_ _ _ _ _|   |_ _ _ _ _ _|
.
    1      4        8          15           21             33         (End)

Hardy and Wright, "An introduction to the theory of numbers", Oxford University Press, fifth edition, p. 266.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Vaclav Kotesovec, Plot of (a(n) - Pi^2*n^2/12) / (n*log(n)^(2/3)) for n = 2..100000.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916.
Peter Polm, C# program for A024916.
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.

Partial sums of A000203.
Cf. A056550, A104471(2*n-1, n), A123229, A128489, A000217, A134867, A072692, A158905, A237593, A245092, A006218, A222548, A092406, A160664.
Cf. A000385, A010766, A340793.

a024916 n = sum $ map (\k -> k * div n k) [1..n]
-- Reinhard Zumkeller, Apr 20 2015

[(&+[DivisorSigma(1, k): k in [1..n]]): n in [1..60]]; // G. C. Greubel, Mar 15 2019

A024916 := proc(n)
    add(numtheory[sigma](k),k=0..n) ;
end proc: # Zerinvary Lajos, Jan 11 2009
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      numtheory[sigma](n)+a(n-1))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 12 2019

Table[Plus @@ Flatten[Divisors[Range[n]]], {n, 50}] (* Alonso del Arte, Mar 06 2006 *)
Table[Sum[n - Mod[n, m], {m, n}], {n, 50}] (* Roger L. Bagula and Gary W. Adamson, Oct 06 2006 *)
a[n_] := Sum[DivisorSigma[1, k], {k, n}]; Table[a[n], {n, 51}] (* Jean-François Alcover, Dec 16 2011 *)
Accumulate[DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Mar 13 2014 *)

A024916(n)=sum(k=1,n,n\k*k) \\ M. F. Hasler, Nov 22 2007

A024916(z) = { my(s,u,d,n,a,p); s = z*z; u = sqrtint(z); p = 2; for(d=1, u, n = z\d - z\(d+1); if(n<=1, p=d; break(), a = z%d; s -= (2*a+(n-1)*d)*n/2); ); u = z\p; for(d=2, u, s -= z%d); return(s); } \\ See the link for a nicely formatted version. - P. L. Patodia (pannalal(AT)usa.net), Jan 11 2008

A024916(n)={my(s=0,d=1,q=n);while(dPeter Polm, Aug 18 2014

A024916(n)={ my(s=n^2, r=sqrtint(n), nd=n, D); for(d=1, r, (1>=D=nd-nd=n\(d+1)) && (r=d-1) && break; s -= n%d*D+(D-1)*D\2*d); s - sum(d=2, n\(r+1), n%d)} \\ Slightly optimized version of Patodia's code. - M. F. Hasler, Apr 18 2015
(C#) See Polm link.

def A024916(n): return sum(k*(n//k) for k in range(1,n+1)) # Chai Wah Wu, Dec 17 2021

from math import isqrt
def A024916(n): return (-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # Chai Wah Wu, Oct 21 2023

[sum(sigma(k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Mar 15 2019

From Benoit Cloitre, Apr 28 2002: (Start)
a(n) = n^2 - A004125(n).
Asymptotically a(n) = n^2*Pi^2/12 + O(n*log(n)). (End)
G.f.: (1/(1-x))*Sum_{k>=1} x^k/(1-x^k)^2. - Benoit Cloitre, Apr 23 2003
a(n) = Sum_{m=1..n} (n - (n mod m)). - Roger L. Bagula and Gary W. Adamson, Oct 06 2006
a(n) = n^2*Pi^2/12 + O(n*log(n)^(2/3)) [Walfisz]. - Charles R Greathouse IV, Jun 19 2012
a(n) = A000217(n) + A153485(n). - Omar E. Pol, Jan 28 2014
a(n) = A000292(n) - A076664(n), n > 0. - Omar E. Pol, Feb 11 2014
a(n) = A078471(n) + A271342(n). - Omar E. Pol, Apr 08 2016
a(n) = (1/2)*(A222548(n) + A006218(n)). - Ridouane Oudra, Aug 03 2019
From Greg Dresden, Feb 23 2020: (Start)
a(n) = A092406(n) + 8, n>3.
a(n) = A160664(n) - 1, n>0. (End)
a(2*n) = A326123(n) + A326124(n). - Vaclav Kotesovec, Aug 18 2021
a(n) = Sum_{k=1..n} k * A010766(n,k). - Georg Fischer, Mar 04 2022

A018804 Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

Original entry on oeis.org

1, 3, 5, 8, 9, 15, 13, 20, 21, 27, 21, 40, 25, 39, 45, 48, 33, 63, 37, 72, 65, 63, 45, 100, 65, 75, 81, 104, 57, 135, 61, 112, 105, 99, 117, 168, 73, 111, 125, 180, 81, 195, 85, 168, 189, 135, 93, 240, 133, 195, 165, 200, 105, 243, 189, 260, 185, 171, 117, 360

Offset: 1

David W. Wilson

a(n) is the number of times the number 1 appears in the character table of the cyclic group C_n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 02 2001
a(n) is the number of ways to express all fractions f/g whereby each product (f/g)*n is a natural number between 1 and n (using fractions of the form f/g with 1 <= f,g <= n). For example, for n=4 there are 8 such fractions: 1/1, 1/2, 2/2, 3/3, 1/4, 2/4, 3/4 and 4/4. - Ron Lalonde (ronronronlalonde(AT)hotmail.com), Oct 03 2002
Number of non-congruent solutions to xy == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003
Conjecture: n>1 divides a(n)+1 iff n is prime. - Thomas Ordowski, Oct 22 2014
The above conjecture is false, with counterexample given by n = 3*37*43*42307*116341 and a(n)+1 = 26*n. - Varun Vejalla, Jun 19 2025
a(n) is the number of 0's in the multiplication table Z/nZ (cf. A000010 for number of 1's). - Eric Desbiaux, Jun 11 2015
{a(n)} == 1, 3, 1, 0, 1, 3, 1, 0, ... (mod 4). - Isaac Saffold, Dec 30 2017
Since a(p^e) = p^(e-1)*((p-1)e+p) it follows a(p) = 2p-1 and therefore p divides a(p)+1. - Ruediger Jehn, Jun 23 2022

			G.f. = x + 3*x^2 + 5*x^3 + 8*x^4 + 9*x^5 + 15*x^6 + 13*x^7 + 20*x^8 + ...

S. S. Pillai, On an arithmetic function, J. Annamalai University 2 (1933), pp. 243-248.
J. Sándor, A generalized Pillai function, Octogon Mathematical Magazine Vol. 9, No. 2 (2001), 746-748.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 2000 terms from T. D. Noe)
U. Abel, W. Awan, and V. Kushnirevych, A Generalization of the Gcd-Sum Function, J. Int. Seq. 16 (2013) #13.6.7.
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Olivier Bordelles, The Composition of the gcd and Certain Arithmetic Functions, J. Int. Seq. 13 (2010) #10.7.1.
Olivier Bordelles, An Asymptotic Formula for Short Sums of Gcd-Sum Functions, Journal of Integer Sequences, Vol. 15 (2012), #12.6.8.
Olivier Bordelles, A Multidimensional Cesáro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.
Kevin A. Broughan, The gcd-sum function, Journal of Integer Sequences 4 (2001), Article 01.2.2, 19 pp.
J.-M. De Koninck and I. Kátai, Some remarks on a paper of L. Toth, JIS 13 (2010) 10.1.2.
Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
Soichi Ikeda and Kaneaki Matsuoka, On the Lcm-Sum Function, Journal of Integer Sequences, Vol. 17 (2014), Article 14.1.7.
Mathematical Reflections, Solution to Problem O364, Issue 2, 2016, p 24. [Cached copy at Wayback Machine]
Taylor McAdam, Almost-primes in horospherical flows on the space of lattices, arXiv:1802.08764 [math.DS], 2018.
Taylor McAdam, Almost-prime times in horospherical flows on the space of lattices, Journal of Modern Dynamics (2019) Vol. 15, 277-327.
J. Ransford, A Sum of Gcd’s over Friable Numbers, JIS vol 19 (2016) # 16.3.2
Jeffrey Shallit, Problem E 2821, American Mathematical Monthly 87 (1980), 220.
Jeffrey Shallit, Solution, American Mathematical Monthly, 88 (1981), 444-445.
László Tóth, A gcd-sum function over regular integers modulo n, JIS 12 (2009) 09.2.5.
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2.
László Tóth, A survey of gcd-sum functions, J. Integer Sequences 13 (2010), Article 10.8.1, 23 pp.
László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
D. Zhang and W. Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq. 13 (2010), 10.4.7.
D. Zhang and W. Zhai, Mean Values of a Class of Arithmetical Functions, J. Int. Seq. 14 (2011) #11.6.5.
D. Zhang and W. Zhai, On an Open Problem of Tóth, J. Int. Seq. 16 (2013) #13.6.5.

Column 1 of A343510 and A343516.
Cf. A080997, A080998 for rankings of the positive integers in terms of centrality, defined to be the average fraction of an integer that it shares with the other integers as a gcd, or A018804(n)/n^2, also A080999, a permutation of this sequence (A080999(n) = A018804(A080997(n))).
Cf. A185210, A010766, A054523, A127468, A050873, A078430, A095026, A227507, A000010, A344372, A272718 (partial sums), A368736 - A368741.

a018804 n = sum $ map (gcd n) [1..n]  -- Reinhard Zumkeller, Jul 16 2012

[&+[Gcd(n,k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Nov 14 2019

a:=n->sum(igcd(n,j),j=1..n): seq(a(n), n=1..60); # Zerinvary Lajos, Nov 05 2006

f[n_] := Block[{d = Divisors[n]}, Sum[ d*EulerPhi[n/d], {d, d}]]; Table[f[n], {n, 60}] (* Robert G. Wilson v, Mar 20 2012 *)
a[ n_] := If[ n < 1, 0, n Sum[ EulerPhi[d] / d, {d, Divisors@n}]]; (* Michael Somos, Jan 07 2017 *)
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jul 19 2019 *)

{a(n) = direuler(p=2, n, (1 - X) / (1 - p*X)^2)[n]}; /* Michael Somos, May 31 2000 */

a(n)={ my(ct=0); for(i=0,n-1,for(j=0,n-1, ct+=(Mod(i*j,n)==0) ) ); ct; } \\ Joerg Arndt, Aug 03 2013

a(n)=my(f=factor(n)); prod(i=1,#f~,(f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]) \\ Charles R Greathouse IV, Oct 28 2014

a(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ Michel Marcus, Jan 07 2017

from sympy.ntheory import totient, divisors
print([sum(n*totient(d)//d for d in divisors(n)) for n in range(1, 101)]) # Indranil Ghosh, Apr 04 2017

from sympy import factorint
from math import prod
def A018804(n): return prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()) # Chai Wah Wu, Nov 29 2021

a(n) = Sum_{d|n} d*phi(n/d), where phi(n) is Euler totient function (cf. A000010). - Vladeta Jovovic, Apr 04 2001
Multiplicative; for prime p, a(p^e) = p^(e-1)*((p-1)e+p).
Dirichlet g.f.: zeta(s-1)^2/zeta(s).
a(n) = Sum_{d|n} d*tau(d)*mu(n/d). - Benoit Cloitre, Oct 23 2003
Equals A054523 * [1,2,3,...]. Equals row sums of triangle A010766. - Gary W. Adamson, May 20 2007
Equals inverse Mobius transform of A029935 = A054525 * (1, 2, 4, 5, 8, 8, 12, 12, ...). - Gary W. Adamson, Aug 02 2008, corrected Feb 07 2023
Equals row sums of triangle A127478. - Gary W. Adamson, Aug 03 2008
G.f.: Sum_{k>=1} phi(k)*x^k/(1 - x^k)^2, where phi(k) is the Euler totient function. - Ilya Gutkovskiy, Jan 02 2017
a(n) = Sum_{a = 1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1, for n > 1. The sum is over a,b,c such that n*c - a*b = 0. - Benedict W. J. Irwin, Apr 04 2017
Proof: Let gcd(a, n) = g and x = n/g. Define B = {x, 2*x, ..., g*x}; then for all b in B there exists a number c such that a*b = n*c. Since the set B has g elements it follows that Sum_{b=1..n} Sum_{c=1..n} 1 >= g = gcd(a, n) and therefore Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 >= Sum_{a=1..n} gcd(a, n). On the other hand, for all b not in B there is no number c <= n such that a*b = n*c and hence Sum_{b = 1..n} Sum_{c = 1..n} 1 = g. Therefore Sum_{a=1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1 = a(n). - Ruediger Jehn, Jun 23 2022
a(2*n) = a(n)*(3-A007814(n)/(A007814(n)+2)) - Velin Yanev, Jun 30 2017
Proof: Let m = A007814(m) and decompose n into n = k*2^m. We know from Chai Wah Wu's program below that a(n) = Product(p_i^(e_i-1)*((p_i-1)*e_i+p_i)) where the numbers p_i are the prime factors of n and e_i are the corresponding exponents. Hence a(2n) = 2^m*(m+3)*a(k) = 2^m*(m+3)*a(k). On the other hand, a(n) = 2^(m-1)*(m+2)*a(k). Dividing the first equation by the second yields a(2n)/a(n) = 2*(m+3)/(m+2), which equals 3 - m/(m+2). Hence a(2n) = a(n)*(3 - m/(m+2)). - Ruediger Jehn, Jun 23 2022
Sum_{k=1..n} a(k) ~ 3*n^2/Pi^2 * (log(n) - 1/2 + 2*gamma - 6*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 08 2019
a(n) = Sum_{k=1..n} n/gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021
log(a(n)/n) << log n log log log n/log log n; in particular, a(n) << n^(1+e) for any e > 0. See Broughan link for bounds in terms of omega(n). - Charles R Greathouse IV, Sep 08 2022
a(n) = (1/4)*Sum_{k = 1..4*n} (-1)^k * gcd(k, 4*n) = (1/4) * A344372(2*n). - Peter Bala, Jan 01 2024

A126988 Triangle read by rows: T(n,k) = n/k if k is a divisor of n; T(n,k) = 0 if k is not a divisor of n (1 <= k <= n).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Dec 31 2006

Row sums = A000203, sigma(n).
k-th column (k=0,1,2,...) is (1,2,3,...) interspersed with n consecutive zeros starting after the "1".
The nonzero entries of row n are the divisors of n in decreasing order. - Emeric Deutsch, Jan 17 2007
Alternating row sums give A000593. - Omar E. Pol, Feb 11 2018
T(n,k) is the number of k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019

			First few rows of the triangle are:
   1;
   2, 1;
   3, 0, 1;
   4, 2, 0, 1;
   5, 0, 0, 0, 1;
   6, 3, 2, 0, 0, 1;
   7, 0, 0, 0, 0, 0, 1;
   8, 4, 0, 2, 0, 0, 0, 1;
   9, 0, 3, 0, 0, 0, 0, 0, 1;
  10, 5, 0, 0, 2, 0, 0, 0, 0, 1;
  ...
sigma(12) = A000203(n) = 28.
sigma(12) = 28, from 12th row = (12 + 6 + 4 + 3 + 2 + 1), deleting the zeros, from left to right.
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of k's are [6, 3, 2, 0, 0, 1] respectively, equaling the 6th row of triangle. - _Omar E. Pol_, Nov 25 2019

David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, Inc, 2005, Appendix B.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A000203, A008284, A127446, A244051, A328361.

a126988 n k = a126988_tabl !! (n-1) !! (k-1)
a126988_row n = a126988_tabl !! (n-1)
a126988_tabl = zipWith (zipWith (*)) a010766_tabl a051731_tabl
-- Reinhard Zumkeller, Jan 20 2014

[[(n mod k) eq 0 select n/k else 0: k in [1..n]]: n in [1..12]]; // G. C. Greubel, May 29 2019

A126988:=proc(n,k) if type(n/k, integer)=true then n/k else 0 fi end: for n from 1 to 12 do seq(A126988(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 17 2007

Table[If[Mod[n, m]==0, n/m, 0], {n,1,12}, {m,1,n}]//Flatten (* Roger L. Bagula, Sep 06 2008, simplified by Franklin T. Adams-Watters, Aug 24 2011 *)

{T(n,k) = if(n%k==0, n/k, 0)}; \\ G. C. Greubel, May 29 2019

def T(n, k):
    if (n%k==0): return n/k
    else: return 0
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 29 2019

From Emeric Deutsch, Jan 17 2007: (Start)
G.f. of column k: z^k/(1-z^k)^2 (k=1,2,...).
G.f.: G(t,z) = Sum_{k>=1} t^k*z^k/(1-z^k)^2. (End)
G.f.: F(x,z) = log(1/(Product_{n >= 1} (1 - x*z^n))) = Sum_{n >= 1} (x*z)^n/(n*(1 - z^n)) = x*z + (2*x + x^2)*z^2/2 + (3*x + x^3)*z^3/3 + .... Note, exp(F(x,z)) is a g.f. for A008284 (with an additional term T(0,0) equal to 1). - Peter Bala, Jan 13 2015
T(n,k) = A010766(n,k)*A051731(n,k), k=1..n. - Reinhard Zumkeller, Jan 20 2014

Edited by N. J. A. Sloane, Jan 24 2007

Comment from Emeric Deutsch made name by Franklin T. Adams-Watters, Aug 24 2011

A002266 Integers repeated 5 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 0

N. J. A. Sloane

For n > 3, number of consecutive "11's" after the (n+3) "1's" in the continued fraction for sqrt(L(n+2)/L(n)) where L(n) is the n-th Lucas number A000032 (see example). E.g., the continued fraction for sqrt(L(11)/L(9)) is [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 58, 2, 4, 1, ...] with 12 consecutive ones followed by floor(11/5)=2 elevens. - Benoit Cloitre, Jan 08 2006
Complement of A010874, since A010874(n) + 5*a(n) = n. - Hieronymus Fischer, Jun 01 2007
From Paul Curtz, May 13 2020: (Start)
Main N-S vertical of the pentagonal spiral built with this sequence is A001105:
                              21
                          20  15  15
                      20  14  10  10  15
                  20  14   9   6   6  10  15
              20  14   9   5   3   3   6  10  15
          20  14   9   5   2   1   1   3   6  10  16
      19  14   9   5   2   0   0   0   1   3   6  11  16
        19   13   9   5  2   0   0    1   3   7 11  16
          19  13    8  5   2   2   1   4   7  11  16
            19   13   8  4   4   4   4   7  11  16
              19  13   8   8   8   7   7  11  17
                 18  13  12 12  12  12  12  17
                   18  18 18  18  17  17  17
The main S-N vertical and the next one are A000217. (End)

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

a(n) = A010766(n, 5).
Cf. A002162, A008648, A004526, A002264, A002265, A010761, A010762, A110532, A110533, A010872, A010873, A010874.
Partial sums: A130520.

a002266 = (`div` 5)
a002266_list = [0,0,0,0,0] ++ map (+ 1) a002266_list
-- Reinhard Zumkeller, Nov 27 2012

A002266:=n->floor(n/5); seq(A002266(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[n/5], {n,0,20}] (* Wesley Ivan Hurt, Dec 10 2013 *)
Table[{n,n,n,n,n},{n,0,20}]//Flatten (* Harvey P. Dale, Jun 17 2022 *)

a(n)=n\5 \\ Charles R Greathouse IV, Dec 10 2013

def A002266(n): return n//5 # Chai Wah Wu, Nov 08 2022

[floor(n/5) - 1 for n in range(5,88)] # Zerinvary Lajos, Dec 01 2009

a(n) = floor(n/5), n >= 0.
G.f.: x^5/((1-x)(1-x^5)).
a(n) = (n - A010874(n))/5. - Hieronymus Fischer, May 29 2007
For n >= 5, a(n) = floor(log_5(5^a(n-1) + 5^a(n-2) + 5^a(n-3) + 5^a(n-4) + 5^a(n-5))). - Vladimir Shevelev, Jun 22 2010
Sum_{n>=5} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 30 2022

Incorrect formula removed by Ridouane Oudra, Oct 16 2021

A286001 A table of partitions into consecutive parts (see Comments lines for definition).

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 2, 6, 3, 1, 7, 3, 2, 8, 4, 3, 9, 4, 2, 10, 5, 3, 1, 11, 5, 4, 2, 12, 6, 3, 3, 13, 6, 4, 4, 14, 7, 5, 2, 15, 7, 4, 3, 1, 16, 8, 5, 4, 2, 17, 8, 6, 5, 3, 18, 9, 5, 3, 4, 19, 9, 6, 4, 5, 20, 10, 7, 5, 2, 21, 10, 6, 6, 3, 1, 22, 11, 7, 4, 4, 2, 23, 11, 8, 5, 5, 3, 24, 12, 7, 6, 6, 4, 25, 12, 8, 7, 3, 5

Offset: 1

Omar E. Pol, Apr 30 2017

This is a triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists successive blocks of k consecutive terms, where the m-th block starts with m, m>=1, and the first element of column k is in row k*(k+1)/2.
The partitions of n into consecutive parts are represented from the row n up to row A288529(n) as maximum, but in increasing order, exclusively in the columns where the blocks begin.
More precisely, the partition of n into exactly k consecutive parts (if such partition exists) is represented in the column k from the row n up to row n + k - 1 (see examples).
A288772(n) is the minimum number of rows that are required to represent in this table the partitions of all positive integers <= n into consecutive parts.
A288773(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
A288774(n) is the largest positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.
For a theorem related to this table see A286000.

			Triangle begins:
1;
2;
3,   1;
4,   2;
5,   2;
6,   3,  1;
7,   3,  2;
8,   4,  3;
9,   4,  2;
10,  5,  3,  1;
11,  5,  4,  2;
12,  6,  3,  3;
13,  6,  4,  4;
14,  7,  5,  2;
15,  7,  4,  3,  1;
16,  8,  5,  4,  2;
17,  8,  6,  5,  3;
18,  9,  5,  3,  4;
19,  9,  6,  4,  5;
20, 10,  7,  5,  2;
21, 10,  6,  6,  3,  1;
22, 11,  7,  4,  4,  2;
23, 11,  8,  5,  5,  3;
24, 12,  7,  6,  6,  4;
25, 12,  8,  7,  3,  5;
26, 13,  9,  5,  4,  6;
27, 13,  8,  6,  5,  2;
28, 14,  9,  7,  6,  3,  1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:
.   ------------------------------------------------------------------------
Fig:   A     B       C         D          E            F             G
.   ------------------------------------------------------------------------
. n:   1     2       3         4          5            6             7
Row ------------------------------------------------------------------------
1   | [1];|  1; |  1;     |  1;    |  1;        |  1;         |  1;        |
2   |     | [2];|  2;     |  2;    |  2;        |  2;         |  2;        |
3   |     |     | [3],[1];|  3,  1;|  3,  1;    |  3,  1;     |  3,  1;    |
4   |     |     |  4 ,[2];| [4], 2;|  4,  2;    |  4,  2;     |  4,  2;    |
5   |     |     |         |        | [5],[2];   |  5,  2;     |  5,  2;    |
6   |     |     |         |        |  6, [3], 3;| [6], 3, [1];|  6,  3,  1;|
7   |     |     |         |        |            |  7,  3, [2];| [7],[3], 2;|
8   |     |     |         |        |            |  8,  4, [3];|  8, [4], 3;|
.   ------------------------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts (but with the parts in increasing order) are [6] and [1, 2, 3]. These partitions have 1 and 3 consecutive parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 6.
.
Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:
.    --------------------------------------------------------------------
Fig:        H             I                  J                 K
.    --------------------------------------------------------------------
. n:        8             9                  10                11
Row  --------------------------------------------------------------------
1    |  1;        |  1;            |   1;             |   1;            |
1    |  2;        |  2;            |   2;             |   2;            |
3    |  3,  1;    |  3,  1;        |   3,  1;         |   3,  1;        |
4    |  4,  2;    |  4,  2;        |   4,  2;         |   4,  2;        |
5    |  5,  2;    |  5,  2;        |   5,  2;         |   5,  2;        |
6    |  6,  3,  3;|  6,  3,  1;    |   6,  3,  1;     |   6,  3,  1;    |
7    |  7,  3,  2;|  7,  3,  2;    |   7,  3,  2;     |   7,  3,  2;    |
8    | [8], 4,  1;|  8,  4,  3;    |   8,  4,  3;     |   8,  4,  3;    |
9    |            | [9],[4],[2];   |   9,  4,  2;     |   9,  4,  2;    |
10   |            | 10, [5],[3], 1;| [10], 5,  3, [1];|  10,  5,  3,  1;|
11   |            | 11,  5, [4], 2;|  11,  5,  4, [2];| [11],[5], 4,  2;|
12   |            |                |  12,  6,  3, [3];|  12, [6], 3,  3;|
13   |            |                |  13,  6,  4, [4];|  13,  6,  4,  4;|
.    --------------------------------------------------------------------
Figure J: For n = 10 the partitions of 10 into consecutive parts (but with the parts in increasing order) are [10] and [1, 2, 3, 4]. These partitions have 1 and 4 consecutive parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 4 of this table, starting at the row 10.
.
Illustration of initial terms arranged into the diagram of the triangle A237591:
.                                                           _
.                                                         _|1|
.                                                       _|2 _|
.                                                     _|3  |1|
.                                                   _|4   _|2|
.                                                 _|5    |2 _|
.                                               _|6     _|3|1|
.                                             _|7      |3  |2|
.                                           _|8       _|4 _|3|
.                                         _|9        |4  |2 _|
.                                       _|10        _|5  |3|1|
.                                     _|11         |5   _|4|2|
.                                   _|12          _|6  |3  |3|
.                                 _|13           |6    |4 _|4|
.                               _|14            _|7   _|5|2 _|
.                             _|15             |7    |4  |3|1|
.                           _|16              _|8    |5  |4|2|
.                         _|17               |8     _|6 _|5|3|
.                       _|18                _|9    |5  |3  |4|
.                     _|19                 |9      |6  |4 _|5|
.                   _|20                  _|10    _|7  |5|2 _|
.                 _|21                   |10     |6   _|6|3|1|
.               _|22                    _|11     |7  |4  |4|2|
.             _|23                     |11      _|8  |5  |5|3|
.           _|24                      _|12     |7    |6 _|6|4|
.         _|25                       |12       |8   _|7|3  |5|
.       _|26                        _|13      _|9  |5  |4 _|6|
.     _|27                         |13       |8    |6  |5|2 _|
.    |28                           |14       |9    |7  |6|3|1|
...
The number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of partitions of n into consecutive parts.
.
From _Omar E. Pol_, Dec 15 2020: (Start)
The connection (described step by step) between the triangle of A299765 and the above geometric diagram is as follows:
.
   [1];                                       [1];
   [2];                                       [2];
   [3], [2, 1];                               [3], [2, 1];
   [4];                                       [4];
   [5], [3, 2];                               [5], [3, 2];
   [6], [3, 2, 1];                            [6],         [3, 2, 1];
   [7], [4, 3];                               [7], [4, 3];
   [8];                                       [8];
   [9], [5, 4], [4, 3, 2];                    [9], [5, 4], [4, 3, 2];
.
         Figure 1.                                   Figure 2.
.
We start with the irregular                Then we write the same triangle
triangle of A299765 in which               but ordered in columns where the
row n lists the partitions                 column k lists the partitions of
of n into consecutive parts.               n into k consecutive parts.
.
.   _                                          _
    1|                                        |1
    _                                          _
    2|                                        |2
    _    _ _                                   _      _
    3|   2,1|                                 |3     |1
    _                                          _     |2
    4|                                        |4
    _    _ _                                   _      _
    5|   3,2|                                 |5     |2
    _           _ _ _                          _     |3      _
    6|          3,2,1|                        |6            |1
    _    _ _                                   _      _     |2
    7|   4,3|                                 |7     |3     |3
    _                                          _     |4
    8|                                        |8
    _    _ _    _ _ _                          _      _      _
    9|   5,4|   4,3,2|                        |9     |4     |2
                                                     |5     |3
                                                            |4
.
         Figure 3.                                Figure 4.
.
Then we draw to the right of               Then we rotate each sub-diagram
each partition a vertical                  90 degrees counterclockwise.
toothpick and above each part              Every horizontal toothpick represents
we draw a horizontal toothpick.            the existence of that partition.
.                                          The number of vertical toothpicks
.                                          equals the number of parts.
.
.                     _                                      _
                    _|1                                    _|1
                  _|2 _                                  _|2 _
                _|3  |1                                _|3  |1
              _|4   _|2                              _|4   _|2
            _|5    |2 _                            _|5    |2 _
          _|6     _|3|1                          _|6     _|3|1
        _|7      |3  |2                        _|7      |3  |2
      _|8       _|4 _|3                      _|8       _|4 _|3
     |9        |4  |2                       |9        |4  |2
               |5  |3
                   |4
.
         Figure 5.                                Figure 6.
.
Then we join the sub-diagrams              Finally we erase the parts that
forming staircases (or zig-zag             are beyond a certain level (in
paths) that represent the                  this case beyond the 9th level)
partitions that have the same              to make the diagram more standard.
number of parts.
.
The numbers in the k-th staircase (from left to right) are the elements of the k-th column of the triangular array.
Note that this diagram is essentially the same diagram used to represent the triangles A237048, A235791, A237591, and other related sequences such as A001227, A060831 and A204217.
There is an infinite family of this kind of triangles, which are related to polygonal numbers and partitions into consecutive parts that differ by d. For more information see the theorems in A285914 and A303300.
Note that if we take two images of the diagram mirroring each other, with the y-axis in the middle of them, then a new diagram is formed, which is symmetric and represents the sequence A237593 as an isosceles triangle. Then if we fold each level (or row) of that isosceles triangle we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n). (End)

Another version of A286000.
Tables of the same family where the consecutive parts differ by d are A010766 (d=0), this sequence (d=1), A332266 (d=2), A334945 (d=3), A334618(d=4).
Cf. A000217, A001227, A003056, A109814, A196020, A204217, A211343, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A280850, A280851, A285914, A286013, A288529, A288772, A288773, A288774, A296508, A299765, A303300.

A008620 Positive integers repeated three times.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26

Offset: 0

N. J. A. Sloane

Arises from Gleason's theorem on self-dual codes: the Molien series is 1/((1-x^8)*(1-x^24)) for the weight enumerators of doubly-even binary self-dual codes; also 1/((1-x^4)*(1-x^12)) for ternary self-dual codes.
The number of partitions of n into distinct parts where each part is either a power of two or three times a power of two.
Number of partitions of n into parts 1 or 3. - Reinhard Zumkeller, Aug 15 2011
The dimension of the space of modular forms on Gamma_1(3) of weight n>0 with a(q) the generator of weight 1 and c(q)^3 / 27 the generator of weight 3 where a(), c() are cubic AGM theta functions. - Michael Somos, Apr 01 2015
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
a(n-1) is the minimal number of circles that can be drawn through n points in general position in the plane. - Anton Zakharov, Dec 31 2016
Number of partitions of n into distinct parts from A029744.- R. J. Mathar, Mar 01 2023
Number of representations n=sum_i c_i*2^i with c_i in {0,1,3,4} [Anders]. See A120562 or A309025 for other c_i sets. - R. J. Mathar, Mar 01 2023

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 12 Exer. 7
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Eq. (14), p. 601 and Theorem 3c, p. 602; also Problem 5 p. 620.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 1.
E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's Theorem on Self-Dual Codes, IEEE Trans. Information Theory, IT-18 (1972), 409-414.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 210
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 449
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A000027, A001840, A002264, A004016, A004396, A005882, A005928, A008621, A010766, A049347.

Column 3 of A235791.

a008620 = (+ 1) . (`div` 3)
a008620_list = concatMap (replicate 3) [1..]
-- Reinhard Zumkeller, Feb 19 2013, Apr 16 2012, Sep 25 2011

[Floor(n/3)+1: n in [0..80]]; // Vincenzo Librandi, Aug 16 2011

a := func< n | Dimension( ModularForms( Gamma1(3), n))>; /* Michael Somos, Apr 01 2015 */

A008620:=n->floor(n/3)+1; seq(A008620(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Table[Floor[n/3] + 1, {n, 0, 90}] (* Stefan Steinerberger, Apr 02 2006 *)
Table[{n, n, n}, {n, 30}] // Flatten (* Harvey P. Dale, Jan 15 2017 *)
Ceiling[Range[20]/3] (* Eric W. Weisstein, Aug 12 2023 *)
Table[Ceiling[n/3], {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(1 + n - Cos[2 n Pi/3] + Sin[2 n Pi/3]/Sqrt[3])/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - ChebyshevU[n, -1/2] + 1)/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 1, 2}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[1/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)

PARI
```
a(n)=n\3+1
```

def a(n) : return( dimension_modular_forms( Gamma1(3), n) ); # Michael Somos, Apr 01 2015

a(n) = floor(n/3) + 1.
a(n) = A010766(n+3, 3).
G.f.: 1/((1-x)*(1-x^3)) = 1/((1-x)^2*(1+x+x^2)).
a(n) = A001840(n+1) - A001840(n). - Reinhard Zumkeller, Aug 01 2002
From Paul Barry, May 19 2004: (Start)
Convolution of A049347 and A000027.
a(n) = Sum_{k=0..n} (k+1)*2*sqrt(3)*cos(2*Pi*(n-k)/3 + Pi/6)/3. (End)
The g.f. is 1/(1-V_trefoil(x)), where V_trefoil is the Jones polynomial of the trefoil knot. - Paul Barry, Oct 08 2004
a(2n) = A004396(n+1). - Philippe Deléham, Dec 14 2006
a(n) = ceiling(n/3), n>=1. - Mohammad K. Azarian, May 22 2007
E.g.f.: exp(x)*(2 + x)/3 + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022

A008621 Expansion of 1/((1-x)*(1-x^4)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21

Offset: 0

N. J. A. Sloane

Arises from Gleason's theorem on self-dual codes: 1/((1-x^2)*(1-x^8)) is the Molien series for the real 2-dimensional Clifford group (a dihedral group of order 16) of genus 1.
Thickness of the hypercube graph Q_n. - Eric W. Weisstein, Sep 09 2008
Count of odd numbers between consecutive quarter-squares, A002620. Oppermann's conjecture states that for each count there will be at least one prime. - Fred Daniel Kline, Sep 10 2011
Number of partitions into parts 1 and 4. - Joerg Arndt, Jun 01 2013
a(n-1) is the minimum independence number over all planar graphs with n vertices. The bound follows from the Four Color Theorem. It is attained by a union of 4-cliques. Other extremal graphs are examined in the Bickle link. - Allan Bickle, Feb 04 2022

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Problem 3, p. 602.

T. D. Noe, Table of n, a(n) for n = 0..1000
Allan Bickle, Independence Number of Maximal Planar Graphs, Congr. Num. 234 (2019) 61-68.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 211
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Eric Weisstein's World of Mathematics, Graph Thickness
Wikipedia, Oppermann's conjecture
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002265 (equals this - 1).

Cf. A008718, A024186, A110160, A110868, A110869, A110876, A110880, A008620.

Table[Floor[n/4]+1, {n, 0, 80}] (* Stefan Steinerberger, Apr 03 2006 *)
CoefficientList[Series[1/((1-x)(1-x^4)),{x,0,80}],x] (* Harvey P. Dale, Feb 19 2012 *)
Flatten[ Table[ PadRight[{},4,n],{n,19}]] (* Harvey P. Dale, Feb 19 2012 *)

a(n)=n\4+1 \\ Charles R Greathouse IV, Feb 06 2017

[n//4+1 for n in range(85)] # Gennady Eremin, Mar 01 2022

a(n) = floor(n/4) + 1.
a(n) = A010766(n+4, 4).
Also, a(n) = ceiling((n+1)/4), n >= 0. - Mohammad K. Azarian, May 22 2007
a(n) = Sum_{i=0..n} A121262(i) = n/4 + 5/8 + (-1)^n/8 + A057077(n)/4. - R. J. Mathar, Mar 14 2011
a(x,y) := floor(x/2) + floor(y/2) - x where x = A002620(n) and y = A002620(n+1), n > 2. - Fred Daniel Kline, Sep 10 2011
a(n) = a(n-1) + a(n-4) - a(n-5); a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=2. - Harvey P. Dale, Feb 19 2012
From R. J. Mathar, Jun 04 2021: (Start)
G.f.: 1 / ( (1+x)*(1+x^2)*(x-1)^2 ).
a(n) + a(n-1) = A004524(n+3).
a(n) + a(n-2) = A008619(n). (End)
a(n) = A002265(n) + 1. - M. F. Hasler, Oct 17 2022

More terms from Stefan Steinerberger, Apr 03 2006

A048158 Triangular array T read by rows: T(n,k) = n mod k, for k=1,2,...,n, n=1,2,...

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 3, 2, 1, 0, 0, 0, 2, 0, 3, 2, 1, 0, 0, 1, 0, 1, 4, 3, 2, 1, 0, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 2, 0, 5, 4, 3, 2, 1, 0, 0, 1, 1, 1, 3, 1, 6, 5, 4, 3, 2, 1, 0, 0, 0, 2, 2, 4, 2, 0, 6, 5, 4, 3, 2, 1, 0

Offset: 1

Clark Kimberling

Also, rectangular array read by antidiagonals: a(n, k) = n mod k, n >= 0, k >= 1. Cf. A051126, A051127, A051777. - David Wasserman, Oct 01 2008

			Triangle begins
  0;
  0  0;
  0  1  0;
  0  0  1  0;
  0  1  2  1  0;
  0  0  0  2  1  0;
  0  1  1  3  2  1  0;
  0  0  2  0  3  2  1  0;
  0  1  0  1  4  3  2  1  0;
  0  0  1  2  0  4  3  2  1  0;
  0  1  2  3  1  5  4  3  2  1  0;
  0  0  0  0  2  0  5  4  3  2  1  0;
  ...
From _Omar E. Pol_, Feb 21 2014: (Start)
Illustration of the 12th row of triangle:
-----------------------------------
.      k: 1 2 3 4 5 6 7 8 9 10..12
-----------------------------------
.         _ _ _ _ _ _ _ _ _ _ _ _
.        |_| | | | | | | | | | | |
.        |_|_| | | | | | | | | | |
.        |_| |_| | | | | | | | | |
.        |_|_| |_| | | | | | | | |
.        |_| | | |_| | | | | | | |
.        |_|_|_| | |_| | | | | | |
.        |_| | | | | |_| | | | | |
.        |_|_| |_| | |*|_| | | | |
.        |_| |_| | | |* *|_| | | |
.        |_|_| | |_| |* * *|_| | |
.        |_| | | |*| |* * * *|_| |
.        |_|_|_|_|*|_|* * * * *|_|
.
Row 12 is 0 0 0 0 2 0 5 4 3 2 1 0
(End)

Alois P. Heinz, Rows n = 1..141, flattened
Michael Z. Spivey, The Humble Sum of Remainders Function, Mathematics Magazine, Vol. 78, No. 4 (Oct., 2005), pp. 300-305.
Eric Weisstein's World of Mathematics, Mod.

Row sums are given by A004125.
Cf. A002260.
Cf. A000007, A010766, A051126, A051127, A051731, A051777.

a048158 = mod
a048158_row n = a048158_tabl !! (n-1)
a048158_tabl = zipWith (map . mod) [1..] a002260_tabl
-- Reinhard Zumkeller, Apr 29 2015, Jan 20 2014 (fixed), Aug 13 2013

T:= (n, k)-> modp(n, k):
seq(seq(T(n, k), k=1..n), n=1..20); # Alois P. Heinz, Apr 04 2012

Flatten[Table[Mod[n, Range[n]], {n, 15}]]

def A048158_T(n,k): return n%k # Chai Wah Wu, May 13 2024

A051731(n,k) = A000007(T(n,k)). - Reinhard Zumkeller, Nov 01 2009
T(n,k) = n - k*A010766(n,k). - Mats Granvik, Gary W. Adamson, Feb 20 2010
G.f. for the k-th column: x^(k+1)*Sum_{i=0..k-2} (i + 1)*x^i/(1 - x^k). - Stefano Spezia, May 08 2024

More terms from David Wasserman, Oct 01 2008

cp's OEIS Frontend

A008619 Positive integers repeated.

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A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

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A018804 Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

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A126988 Triangle read by rows: T(n,k) = n/k if k is a divisor of n; T(n,k) = 0 if k is not a divisor of n (1 <= k <= n).

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