A244051 -id:A244051 - cp's OEIS frontend

A027750 Triangle read by rows in which row n lists the divisors of n.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 29

Offset: 1

N. J. A. Sloane

Or, in the list of natural numbers (A000027), replace n with its divisors.
This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.
Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008
Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011
{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012
Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012
Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013
The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015
The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017
T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019
Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025
All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

			Triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 3, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 3, 4, 6, 12;
  ...
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 parts are [1, 2, 3, 6] respectively, the same as the divisors of 6. - _Omar E. Pol_, Nov 20 2019

Franklin T. Adams-Watters, Rows 1..1000, flattened
Franklin T. Adams-Watters, Rows 1..10000
Peter Luschny, A Python program using gcd for n >= 0.
Omar E. Pol, Illustration of initial terms, (2009).
Eric Weisstein's World of Mathematics, Divisor
Wikipedia, Table of divisors
Index entries for sequences related to divisors of numbers

Cf. A000005 (row length), A001221, A027749, A027751, A056534, A056538, A127093, A135010, A161700, A163280, A240698 (partial sums of rows), A240694 (partial products of rows), A247795 (parities), A292226, A244051.

a027750 n k = a027750_row n !! (k-1)
a027750_row n = filter ((== 0) . (mod n)) [1..n]
a027750_tabf = map a027750_row [1..]
-- Reinhard Zumkeller, Jan 15 2011, Oct 21 2010

Magma
```
[Divisors(n) : n in [1..20]];
```

seq(op(numtheory:-divisors(a)), a = 1 .. 20) # Matt C. Anderson, May 15 2017

Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]

v=List();for(n=1,20,fordiv(n,d,listput(v,d)));Vec(v) \\ Charles R Greathouse IV, Apr 28 2011

from sympy import divisors
for n in range(1, 16):
    print(divisors(n)) # Indranil Ghosh, Mar 30 2017

a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - Reinhard Zumkeller, May 10 2006

T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A002131 Sum of divisors d of n such that n/d is odd.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 8, 8, 13, 12, 12, 16, 14, 16, 24, 16, 18, 26, 20, 24, 32, 24, 24, 32, 31, 28, 40, 32, 30, 48, 32, 32, 48, 36, 48, 52, 38, 40, 56, 48, 42, 64, 44, 48, 78, 48, 48, 64, 57, 62, 72, 56, 54, 80, 72, 64, 80, 60, 60, 96, 62, 64, 104, 64, 84, 96, 68, 72, 96, 96, 72

Offset: 1

N. J. A. Sloane, Simon Plouffe

Glaisher calls this Delta'(n) or Delta'1(n). - _N. J. A. Sloane, Nov 24 2018
Equals row sums of triangle A143119. - Gary W. Adamson, Jul 26 2008
Cayley begins article 386 with "To find the value of A, = 8{q/(1-q)^2 + q^3/(1-q^3)^2 +&c.}," where A is 8 time the g.f. of this sequence. - Michael Somos, Aug 01 2011
a(n) = 2*(a(n-1) - a(n-4) + a(n-9) ... +- a(n-i^2) ...) up to the last positive number n - i^2, and if n is a square, then a(0) should be replaced by n/2 (cf. Halphen). - Michel Marcus, Oct 14 2012
From Omar E. Pol, Nov 26 2019: (Start)
a(n) is also the total number of odd parts in the partitions of n into equal parts.
a(n) = n iff n is a power of 2.
a(n) = n + 1 iff n is an odd prime. (End)

			G.f. = q + 2*q^2 + 4*q^3 + 4*q^4 + 6*q^5 + 8*q^6 + 8*q^7 + 8*q^8 + 13*q^9 + ...
The divisors of 6 are 1, 2, 3, and 6. Only 6/2 and 6/6 are odd. Hence, a(6) = 2 + 6 = 8.
As 120 = 15 * 2^3 where 15 is odd and 2^3 is the largest power of 2 dividing 120, a(120) = sigma(15) * 2^3 = 24 * 8 = 192. - _David A. Corneth_, Aug 12 2019
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1]. There are 8 odd parts, so a(6) = 8. - _Omar E. Pol_, Nov 26 2019

A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 294, Art. 386.
G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 346 Exercise XXI(18). MR0121327 (22 #12066)
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eqs. (5.1.29.3), (5.1.29.9).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Heng Huat Chan and Christian Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
G.-H. Halphen, Sur les sommes des diviseurs des nombres entiers et les décompositions en deux carrés, Bull. math. Soc. France, 6 (1877-1878), 119-120.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025. See p. 2.
Index entries for sequences mentioned by Glaisher

A diagonal of A060047. Bisection A008438.

Cf. A000203, A000593, A006519, A026741, A143119, A192065, A244051, A301798, A326938 (Dirichlet inverse).

a002131 n = sum [d | d <- [1..n], mod n d == 0, odd $ div n d]
-- Reinhard Zumkeller, Aug 14 2011

[&+[d:d in Divisors(m)|IsOdd(Floor(m/d))] :m in [1..75]]; // Marius A. Burtea, Aug 12 2019

a:= proc(n) local e;
  e:= 2^padic:-ordp(n,2);
  e*numtheory:-sigma(n/e)
end proc:
map(a, [$1..100]); # Robert Israel, Jul 05 2016

a[n_]:=Total[Cases[Divisors[n], d_ /; OddQ[n/d]]]; Table[a[n],{n,1,71}] (* Jean-François Alcover, Mar 18 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, # / GCD[#, 2] &]] (* Michael Somos, Aug 01 2011 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1/8) EllipticK[ m] ( EllipticK[ m] - EllipticE[ m] ) / (Pi/2 )^2, {q, 0, n}]] (* Michael Somos, Aug 01 2011 *)
Table[Total[Select[Divisors[n],OddQ[n/#]&]],{n,80}] (* Harvey P. Dale, Jun 05 2015 *)
a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ[q]}, (1/2) (EllipticK[ m] / Pi)^2 (D[ JacobiZeta[ JacobiAmplitude[x, m], m], x] /. x -> 0)], {q, 0, n}]; (* Michael Somos, Mar 17 2017 *)
f[2, e_] := 2^e; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

{a(n) = if( n<1, 0, direuler( p=2, n, (1 - (p<3) * X) / ((1 - X) * (1 - p*X))) [n])}; /* Michael Somos, Apr 05 2003 */

{a(n) = if( n<1, 0, sumdiv( n, d, d / gcd(d, 2)))}; /* Michael Somos, Apr 05 2003 */

a(n) = my(v = valuation(n, 2)); sigma(n>>v)<David A. Corneth, Aug 12 2019

from math import prod
from sympy import factorint
def A002131(n): return prod(p**e if p == 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Dec 17 2021

Expansion of K(k^2) * (K(k^2) - E(k^2)) / (2 * Pi^2) in powers of q where q is Jacobi's nome and K(), E() are complete elliptic integrals. - Michael Somos, Aug 01 2011
Multiplicative with a(p^e) = p^e if p = 2; (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Aug 01 2001
a(n) = sigma(n) - sigma(n/2) for even n and = sigma(n) otherwise where sigma(n) is the sum of divisors of n (A000203). - Valery A. Liskovets, Apr 07 2002
G.f.: A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = 2*u1*u6 - u1*u3 - 10*u2*u6 + u2^2 + 2*u2*u3 + 9*u6^2. - Michael Somos, Apr 10 2005
G.f.: A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 - 3*u6)^2 - (u1 - 2*u2) * (u3 - 2*u6). - Michael Somos, Sep 06 2005
G.f.: Sum_{n>=1} n*x^n/(1-x^(2*n)). - Vladeta Jovovic, Oct 16 2002
G.f.: Sum_{k>0} x^(2*k - 1) / (1 - x^(2*k - 1))^2. - Michael Somos, Aug 17 2005
G.f.: (1/8) * theta_4''(0) / theta_4(0) = (Sum_{k>0} -(-1)^k * k^2 q^(k^2)) / (Sum_{k in Z} (-1)^k * q^(k^2)) where theta_4(u) is one of Jacobi's theta functions.
G.f.: A(q) = Z'(0) * K^2 / (2 * Pi^2) = (K - E) * K /(2 * Pi^2) where Z(u) is the Jacobi Zeta function and K, E are complete elliptic integrals. - Michael Somos, Sep 06 2005
Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 1/2^s). - Michael Somos, Apr 05 2003
Moebius transform is A026741.
a(n) = n * Sum_{c|n} 1/c, where c are odd numbers (A005408) dividing n. a(n) = A069359(n) + n. a(n) = A000035(n) (*) A000027(n), where operation (*) denotes Dirichlet convolution, that is, convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d) = Sum_{d|n} A000035(d) * A000027(n/d). -Jaroslav Krizek, Nov 07 2013
L.g.f.: Sum_{ k>0 } atanh(x^k) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
a(n) = A006519(n)*A000203(n/A006519(n)). - Robert Israel, Jul 05 2016
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 /16. - Vaclav Kotesovec, Feb 01 2019
a(n) = (A000203(n) + A000593(n))/2. - Amiram Eldar, Aug 12 2019
From Peter Bala, Jan 06 2021: (Start)
G.f.: A(x) = (1/2)*Sum_{n = -oo..oo} x^(2*n+1)/(1 - x^(2*n+1))^2.
A(x) = Sum_{n = -oo..oo} x^(4*n+1)/(1 - x^(4*n+1))^2.
a(2*n) = 2*a(n); a(2*n+1) = A008438(n). (End)
Expansion of (-1/2) x (d phi(-x) / dx) / phi(-x) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Jul 01 2023

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

A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

Original entry on oeis.org

0, 1, 2, 5, 4, 12, 6, 17, 14, 22, 10, 44, 12, 32, 36, 49, 16, 69, 18, 78, 52, 52, 22, 132, 44, 62, 68, 112, 28, 168, 30, 129, 84, 82, 92, 233, 36, 92, 100, 230, 40, 240, 42, 180, 192, 112, 46, 356, 90, 207, 132, 214, 52, 312, 148, 328, 148, 142, 58, 552, 60

Offset: 1

Labos Elemer, May 28 2004

Not all values arise and some arise more than once.
Row sums of triangle A134866. - Gary W. Adamson, Nov 14 2007
Sum of the largest parts of the partitions of n into two parts such that the smaller part divides the larger. - Wesley Ivan Hurt, Dec 21 2017
a(n) is also the sum of all parts minus the total number of parts of all partitions of n into equal parts (an interpretation of the Torlach Rush's formula). - Omar E. Pol, Nov 30 2019
If and only if sigma(n) divides a(n), then n is one of Ore's Harmonic numbers, A001599. - Antti Karttunen, Jul 18 2020

			q^2 + 2*q^3 + 5*q^4 + 4*q^5 + 12*q^6 + 6*q^7 + 17*q^8 + 14*q^9 + ...
For n = 4 the partitions of 4 into equal parts are [4], [2,2], [1,1,1,1]. The sum of all parts is 4 + 2 + 2 + 1 + 1 + 1 + 1 = 12. There are 7 parts, so a(4) = 12 - 7 = 5. - _Omar E. Pol_, Nov 30 2019

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 30.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A001599, A038040, A134866, A152211, A244051, A324121 (= gcd(a(n), sigma(n))).

Cf. A088827 (positions of odd terms).

using AbstractAlgebra
function A094471(n)
    sum(k for k in 0:n if is_divisible_by(n, n - k))
end
[A094471(n) for n in 1:61] |> println  # Peter Luschny, Nov 14 2023

with(numtheory); A094471:=n->n*tau(n)-sigma(n); seq(A094471(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013
divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0):
a := n -> local k; add(`if`(divides(n - k, n), k, 0), k = 0..n):
seq(a(n), n = 1..61);  # Peter Luschny, Nov 14 2023

Table[n*DivisorSigma[0, n] - DivisorSigma[1, n], {n, 1, 100}]

{a(n) = n*numdiv(n) - sigma(n)} /* Michael Somos, Jan 25 2008 */

from math import prod
from sympy import factorint
def A094471(n):
    f = factorint(n).items()
    return n*prod(e+1 for p,e in f)-prod((p**(e+1)-1)//(p-1) for p,e in f)
# Chai Wah Wu, Nov 14 2023

def A094471(n): return sum(k for k in (0..n) if (n-k).divides(n))
print([A094471(n) for n in range(1, 62)])  # Peter Luschny, Nov 14 2023

a(n) = n*tau(n) - sigma(n) = n*A000005(n) - A000203(n). [Previous name.]
If p is prime, then a(p) = p*tau(p)-sigma(p) = 2p-(p+1) = p-1 = phi(p).
If n>1, then a(n)>0.
a(n) = Sum_{d|n} (n-d). - Amarnath Murthy, Jul 31 2005
G.f.: Sum_{k>=1} k*x^(2*k)/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 24 2018
a(n) = A038040(n) - A000203(n). - Torlach Rush, Feb 02 2019

Simpler name by Peter Luschny, Nov 14 2023

A127446 Triangle T(n,k) = n*A051731(n,k) read by rows.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jan 14 2007

Replace the 1's in row n of A051731 with n's.

T(n,k) is the sum of the k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019

			First few rows of the triangle:
  1;
  2, 2;
  3, 0, 3;
  4, 4, 0, 4;
  5, 0, 0, 0, 5;
  6, 6, 6, 0, 0, 6;
  7, 0, 0, 0, 0, 0, 7;
  ...
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 sum of the k's are [6, 6, 6, 0, 0, 6] respectively, equaling the 6th row of triangle. - _Omar E. Pol_, Nov 25 2019

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

Cf. A038040 (row sums), A051731, A126988, A244051, A328362.

a127446 n k = a127446_tabl !! (n-1) !! (k-1)
a127446_row n = a127446_tabl !! (n-1)
a127446_tabl = zipWith (\v ws -> map (* v) ws) [1..] a051731_tabl
-- Reinhard Zumkeller, Jan 21 2014

A127446 := proc(n,k) if n mod k = 0 then n; else 0; fi; end: for n from 1 to 20 do for k from 1 to n do printf("%d,",A127446(n,k)) ; od: od: # R. J. Mathar, May 08 2009

Flatten[Table[If[Mod[n, k] == 0, n, 0], {n, 20}, {k, n}]] (* Vincenzo Librandi, Nov 02 2016 *)

T(n,k) = k*A126988(n,k). - Omar E. Pol, Nov 25 2019

Edited and extended by R. J. Mathar, May 08 2009

A152211 a(n) = n * sigma_0(n) + sigma_1(n).

Original entry on oeis.org

2, 7, 10, 19, 16, 36, 22, 47, 40, 58, 34, 100, 40, 80, 84, 111, 52, 147, 58, 162, 116, 124, 70, 252, 106, 146, 148, 224, 88, 312, 94, 255, 180, 190, 188, 415, 112, 212, 212, 410, 124, 432, 130, 348, 348, 256, 142, 604

Offset: 1

Ctibor O. Zizka, Nov 29 2008

a(n) is the sum of all parts plus the total number of parts of all partitions of n into equal parts. - Omar E. Pol, Nov 30 2019

			For n = 4 the partitions of 4 into equal parts are [4], [2,2], [1,1,1,1]. The sum of all parts is 4 + 2 + 2 + 1 + 1 + 1 + 1 = 12. There are 7 parts, so a(4) = 12 + 7 = 19. - _Omar E. Pol_, Nov 30 2019

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

Cf. A000005, A000203, A038040, A094471, A244051.

Cf. A001620, A013661.

Array[Total[{#, 1} DivisorSigma[{0, 1}, #]] &, 48] (* Michael De Vlieger, Dec 01 2019 *)

a(n) =  n*numdiv(n) + sigma(n) \\ Michel Marcus, Jun 02 2013

a(n) = n * A000005(n) + A000203(n).
a(n) = A038040(n) + A000203(n). - Torlach Rush, Feb 01 2019
G.f.: Sum_{k>=1} (k + 1) * x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Aug 14 2019
Sum_{k=1..n} a(k) ~ (n^2/2) * (log(n) + 2*gamma + zeta(2) - 1/2), where gamma is Euler's constant (A001620). - Amiram Eldar, Feb 01 2025

A309732 Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.

Original entry on oeis.org

1, 7, 15, 38, 40, 108, 77, 188, 180, 290, 187, 600, 260, 560, 630, 888, 442, 1323, 551, 1620, 1218, 1364, 805, 3024, 1325, 1898, 1998, 3136, 1276, 4680, 1457, 4080, 2970, 3230, 3290, 7470, 2072, 4028, 4134, 8200, 2542, 9072, 2795, 7656, 7830, 5888, 3337, 14496, 4998, 9825, 7038

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

nonn

Dirichlet convolution of triangular numbers (A000217) with squares (A000290).

a(n) is n times half m, where m is the sum of all parts plus the total number of parts of the partitions of n into equal parts. - Omar E. Pol, Nov 30 2019

Marius A. Burtea, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A000217, A000290, A007437, A034714, A034715, A038040, A064987, A152211, A309731, A244051.

[n*(n*NumberOfDivisors(n) + DivisorSigma(1,n))/2:n in [1..51]]; // Marius A. Burtea, Nov 29 2019

with(numtheory): seq(n*(n*tau(n)+sigma(n))/2, n=1..50); # Ridouane Oudra, Nov 28 2019

nmax = 51; CoefficientList[Series[Sum[k^2 x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j (j + 1)/2, j^2, j, n], {n, 1, 51}]
Table[n (n DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 51}]

a(n)=sumdiv(n, d, binomial(n/d+1,2)*d^2); \\ Andrew Howroyd, Aug 14 2019

a(n)=n*(n*numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) = n * (n * d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-2) * (zeta(s-2) + zeta(s-1))/2.
a(n) = n*(A038040(n) + A000203(n))/2 = n*A152211(n)/2. - Omar E. Pol, Aug 17 2019
a(n) = Sum_{k=1..n} k*sigma(gcd(n,k)). - Ridouane Oudra, Nov 28 2019

A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 4, 0, 1, 7, 3, 0, 1, 6, 0, 0, 0, 1, 12, 4, 3, 0, 0, 1, 8, 0, 0, 0, 0, 0, 1, 15, 7, 0, 3, 0, 0, 0, 1, 13, 0, 4, 0, 0, 0, 0, 0, 1, 18, 6, 0, 0, 3, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 24, 8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jun 03 2007

The original definition was: A127093 * A125093^(-1).
Left border = A000203, sigma(n): (1, 3, 4, 7, 6, ...). Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, ...); = inverse Moebius transform applied to sigma(n); (i.e., inverse Moebius transform applied twice to natural numbers).
T(n,k) is the total number of parts congruent to 0 mod k in the partitions of n into equal parts. - Omar E. Pol, Nov 19 2019
From Omar E. Pol, Jan 01 2020: (Start)
Conjecture 1: the sum of odd-indexed terms in row n equals A327096(n).
Conjecture 2: the sum of even-indexed terms in row n equals the n-th term of the sequence formed by A000004 and A007429 interleaved.
Conjecture 3: alternating row sums give A288417. (End)

			First few rows of the triangle are:
   1;
   3,  1;
   4,  0, 1;
   7,  3, 0, 1;
   6,  0, 0, 0, 1;
  12,  4, 3, 0, 0, 1;
   8,  0, 0, 0, 0, 0, 1;
  15,  7, 0, 3, 0, 0, 0, 1;
  13,  0, 4, 0, 0, 0, 0, 0, 1;
  18,  6, 0, 0, 3, 0, 0, 0, 0, 1;
  12,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1;
  14,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  24,  8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1;
  24,  0, 6, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  31, 15, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1;
...
Extended by _Omar E. Pol_, Nov 19 2019

Cf. A000203, A007429, A127093, A125093, A244051, A288417, A327096.

A127093 * A125093^(-1), as infinite lower triangular matrices.

New name and more terms from Omar E. Pol, Nov 19 2019

A329323 Triangle read by rows: T(n,k) is the sum of the parts congruent to 0 mod k in the partitions of n into equal parts, 1 <= k <= n.

Original entry on oeis.org

1, 4, 2, 6, 0, 3, 12, 8, 0, 4, 10, 0, 0, 0, 5, 24, 12, 12, 0, 0, 6, 14, 0, 0, 0, 0, 0, 7, 32, 24, 0, 16, 0, 0, 0, 8, 27, 0, 18, 0, 0, 0, 0, 0, 9, 40, 20, 0, 0, 20, 0, 0, 0, 0, 10, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 72, 48, 36, 24, 0, 24, 0, 0, 0, 0, 0, 12, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 56, 28, 0, 0

Offset: 1

Omar E. Pol, Nov 21 2019

Column k lists the terms of A038040 multiplied by k and interspersed with (k-1) zeros.

			Triangle begins:
   1;
   4,  2;
   6,  0,  3;
  12,  8,  0,  4;
  10,  0,  0,  0,  5;
  24, 12, 12,  0,  0,  6;
  14,  0,  0,  0,  0,  0,  7;
  32, 24,  0, 16,  0,  0,  0,  8;
  27,  0, 18,  0,  0,  0,  0,  0,  9;
  40, 20,  0,  0, 20,  0,  0,  0,  0, 10;
  22,  0,  0,  0,  0,  0,  0,  0,  0,  0, 11;
  72, 48, 36, 24,  0, 24,  0,  0,  0,  0,  0, 12;
  26,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 13;
  56, 28,  0,  0,  0,  0, 28,  0,  0,  0,  0,  0,  0, 14;
  60,  0, 30,  0, 30,  0,  0,  0,  0,  0,  0,  0,  0,  0, 15;
  80, 64,  0, 48,  0,  0,  0, 32,  0,  0,  0,  0,  0,  0,  0, 16;
...
For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. Then, for k = 2 the sum of the parts that are multiples of 2 is 6 + 2 + 2 + 2 = 12, so T(6,2) = 12.

Column 1 is A038040.
Row sums give A034718.
Leading diagonal gives A000027.
The number of positive terms in row n is A000005(n).
Cf. A126988, A130540, A134577, A244051.

T(n,k) = A126988(n,k)*A134577(n,k).

A309400 Irregular triangle read by rows in which row n lists in reverse order the partitions of n into equal parts.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 30 2019

The number of parts in row n equals sigma(n) = A000203(n), the sum of the divisors of n. More generally, the number of parts congruent to 0 (mod m) in row m*n equals sigma(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of the aliquot parts of n.
The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.
The number of partitions in row n equals A000005(n), the number of divisors of n.
The number of partitions in row n with an odd number of parts equals A001227(n).
The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).
The sum of row n equals n*A000005(n) = A038040(n).
Records in row n give the n-th row of A027750.
First n rows contain A000217(n) 1's.
The number of k's in row n is A126988(n,k).
The number of odd parts in row n is A002131(n).
The k-th block in row n has A056538(n,k) parts.
Column 1 gives A000012.
Right border gives A000027.

			Triangle begins:
[1];
[1,1], [2];
[1,1,1], [3];
[1,1,1,1], [2,2], [4];
[1,1,1,1,1], [5];
[1,1,1,1,1,1], [2,2,2], [3,3], [6];
[1,1,1,1,1,1,1], [7];
[1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8];
[1,1,1,1,1,1,1,1,1], [3,3,3], [9];
[1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10];
[1,1,1,1,1,1,1,1,1,1,1], [11];
[1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [12];
[1,1,1,1,1,1,1,1,1,1,1,1,1], [13];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2], [7,7], [14];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [3,3,3,3,3], [5,5,5], [15];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2,2], [4,4,4,4], [8,8], [16];
...

Mirror of A244051.

Cf. A000005, A000012, A000027, A000203, A000217, A001065, A001227, A002131, A027750, A038040, A048050, A056538, A126988, A237593, A245579, A299765, A328365.

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A027750 Triangle read by rows in which row n lists the divisors of n.

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A002131 Sum of divisors d of n such that n/d is odd.

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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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A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

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A127446 Triangle T(n,k) = n*A051731(n,k) read by rows.

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