A008438 -id:A008438 - cp's OEIS frontend

A226255 Number of ways of writing n as the sum of 11 triangular numbers.

1, 11, 55, 176, 440, 957, 1848, 3245, 5412, 8580, 12892, 18888, 26895, 36916, 50160, 66935, 86658, 111870, 142582, 177320, 221100, 272690, 329065, 399102, 480040, 566808, 672969, 793760, 920326, 1074040, 1248412, 1425974, 1640595, 1882145, 2123385, 2418339, 2743928, 3062895, 3453978, 3880855

Offset: 0

N. J. A. Sloane, Jun 01 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.

G.f. is 11th power of g.f. for A010054.
a(0) = 1, a(n) = (11/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 11*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A045823 a(n) = sigma_3(2*n+1).

Original entry on oeis.org

1, 28, 126, 344, 757, 1332, 2198, 3528, 4914, 6860, 9632, 12168, 15751, 20440, 24390, 29792, 37296, 43344, 50654, 61544, 68922, 79508, 95382, 103824, 117993, 137592, 148878, 167832, 192080, 205380, 226982, 260408, 276948, 300764, 340704, 357912

Offset: 0

N. J. A. Sloane

			q + 28*q^3 + 126*q^5 + 344*q^7 + 757*q^9 + 1332*q^11 + 2198*q^13 + ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Equals A045819/2.
Bisection of A001158.
Cf. A008438, A013662, A091986.

[DivisorSigma(3, 2*n+1): n in [0..40]]; // Vincenzo Librandi, Jun 02 2019

A045823 := proc(n)
    numtheory[sigma][3](2*n+1) ;
end proc:
seq(A045823(n),n=0..30) ; # R. J. Mathar, Nov 25 2018

DivisorSigma[3, Range[1, 75, 2]] (* Harvey P. Dale, Jan 11 2015 *)

{a(n) = if( n<0, 0, sigma(2 * n + 1, 3))} /* Michael Somos, Nov 29 2007 */

{a(n) = local(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8) / (2 * eta(x + A)^8 * eta(x^2 + A)^8), n))} /* Michael Somos, Nov 29 2007 */

Expansion of q^(-1) * ( E_4(q) - 9 * E_4(q^2) + 8 * E_4(q^4) ) / 240 in powers of q^2. - Michael Somos, Nov 29 2007
Expansion of q^(-1) * (eta(q^2)^24 + eta(q)^16 * eta(q^4)^8) / (2 * eta(q)^8 * eta(q^2)^8) in powers of q^2. - Michael Somos, Nov 29 2007
a(n) = b(2*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1) if p>2. - Michael Somos, Nov 29 2007
G.f.: (theta_3(q)^8 - theta_4(q)^8) / (32*q) = Sum_{n>=0} sigma_3(2*n+1)*q^(2*n). - Paul D. Hanna, Jun 02 2018
Sum_{k=0..n} a(k) ~ (15*zeta(4)/8) * n^4. - Amiram Eldar, Dec 12 2023

More terms from Benoit Cloitre, Apr 12 2003

A109506 Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 4, -3, 6, -12, 8, -3, 13, -18, 12, -12, 14, -24, 24, -3, 18, -39, 20, -18, 32, -36, 24, -12, 31, -42, 40, -24, 30, -72, 32, -3, 48, -54, 48, -39, 38, -60, 56, -18, 42, -96, 44, -36, 78, -72, 48, -12, 57, -93, 72, -42, 54, -120, 72, -24, 80, -90, 60, -72, 62, -96, 104, -3, 84, -144, 68, -54, 96, -144, 72

Offset: 1

Michael Somos, Jun 30 2005

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Denoted by xi(n) in Glaisher 1907. - Michael Somos, May 17 2013

			q - 3*q^2 + 4*q^3 - 3*q^4 + 6*q^5 - 12*q^6 + 8*q^7 - 3*q^8 + 13*q^9 + ...

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).
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. 8).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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).
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Cf. A000593, A008438, A046897, A096727, A112610.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, -(-1)^n Sum[ If[ Mod[ d, 4] == 0, 0, d], {d, Divisors@n}]] (* Michael Somos, May 17 2013 *)

{a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, if( d%4, d)))}

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); -1/8 * polcoeff( eta(x + A)^8 / eta(x^2 + A)^4, n))}

Expansion of (1 - eta(q)^8 / eta(q^2)^4) / 8 in powers of q.
a(n) = Sum_{d divides n} (-1)^(n/d + d) * d [Glaisher].
Multiplicative with a(2^e) = -3, if e>0. a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f.: Sum_{k>0} k * (x^k / (1 - x^k) - 6 * x^(2*k) / (1 - x^(2*k)) + 8 * x^(4*k) / (1 - x^(4*k))).
G.f.: Sum_{k>0} -(-x)^k / (1 + x^k)^2 = Sum_{k>0} - k * (-x)^k / (1 + x^k).
a(n) = -(-1)^n * A046897(n). a(n) = -A096727(n) / 8 unless n=0. a(2*n) = -3 * A000593(n). a(2*n + 1) = A008438(n). a(4*n + 1) = A112610(n). a(4*n + 3) = A097723(n).
Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023

A247820 Numbers k such that sigma(2*k-1) is a prime p.

Original entry on oeis.org

5, 13, 145, 365, 841, 1201, 1741, 2521, 3961, 5101, 7813, 8581, 13945, 14281, 14965, 41761, 42925, 73345, 139921, 229165, 245701, 265721, 276025, 289561, 298765, 341965, 351961, 353641, 367225, 414961, 595141, 601705, 676285, 697381, 711625, 740545, 942565

Offset: 1

Jaroslav Krizek, Sep 24 2014

nonn

Supersequence of A247789.

Corresponding values of primes p for a(n) are A008438(a(n)-1) or A247837.

			Number 13 is in sequence because sigma(2*13-1) = sigma(25) = 31 (prime).

Martin Møller Skarbiniks Pedersen, Table of n, a(n) for n = 1..1000

Cf. A000203, A008438, A247789, A247837.

[n: n in [1..10000000] | IsPrime(SumOfDivisors(2*n-1))];

for(n=1,10^7,if(isprime(sigma(2*n-1)),print1(n,", "))) \\ Derek Orr, Sep 25 2014

A340949 Number of ways to write n as an ordered sum of 4 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 4, 0, 6, 4, 4, 12, 1, 16, 6, 16, 12, 12, 22, 8, 36, 4, 30, 24, 21, 36, 18, 36, 28, 48, 16, 44, 36, 44, 48, 36, 46, 40, 72, 20, 73, 48, 54, 72, 42, 68, 56, 84, 50, 72, 78, 56, 84, 84, 62, 112, 60, 60, 110, 84, 97, 72, 120, 76, 116, 84, 72, 144, 102, 104, 96, 108, 102, 156, 102, 92

Offset: 4

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A000217, A008438, A010054, A053603, A053604, A319814, A340950, A340951, A340952, A340953, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..75);  # Alois P. Heinz, Jan 31 2021

nmax = 75; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^4, where theta_2() is the Jacobi theta function.

A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 1, 0, 1, 4, 3, 2, 0, 0, 1, 5, 6, 4, 2, 0, 0, 1, 6, 10, 8, 6, 0, 1, 0, 1, 7, 15, 15, 13, 3, 3, 0, 0, 1, 8, 21, 26, 25, 12, 6, 2, 0, 0, 1, 9, 28, 42, 45, 31, 14, 9, 0, 0, 0, 1, 10, 36, 64, 77, 66, 35, 24, 3, 2, 1, 0, 1, 11, 45

Offset: 0

Seiichi Manyama, May 07 2017

A(n, k) is the number of ways of writing n as the sum of k triangular numbers.

			Square array begins:
   1, 1, 1, 1,  1,  1, ...
   0, 1, 2, 3,  4,  5, ...
   0, 0, 1, 3,  6, 10, ...
   0, 1, 2, 4,  8, 15, ...
   0, 0, 2, 6, 13, 25, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.

Main diagonal gives A106337.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* Michael De Vlieger, May 07 2017 *)

G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.

A326123 a(n) is the sum of all divisors of the first n odd numbers.

Original entry on oeis.org

1, 5, 11, 19, 32, 44, 58, 82, 100, 120, 152, 176, 207, 247, 277, 309, 357, 405, 443, 499, 541, 585, 663, 711, 768, 840, 894, 966, 1046, 1106, 1168, 1272, 1356, 1424, 1520, 1592, 1666, 1790, 1886, 1966, 2087, 2171, 2279, 2399, 2489, 2601, 2729, 2849, 2947, 3103, 3205, 3309, 3501, 3609, 3719

Offset: 1

Omar E. Pol, Jun 07 2019

a(n)/A326124(n) converges to 3/5.

a(n) is also the total area of the terraces of the first n odd-indexed levels of the stepped pyramid described in A245092.

			For n = 3 the first three odd numbers are [1, 3, 5] and their divisors are [1], [1, 3], [1, 5] respectively, and the sum of these divisors is 1 + 1 + 3 + 1 + 5 = 11, so a(3) = 11.

Robert Israel, Table of n, a(n) for n = 1..10000

Partial sums of A008438.

Cf. A000203, A005408, A024916, A237593, A245092, A326124.

ListTools:-PartialSums(map(numtheory:-sigma, [seq(i,i=1..200,2)])); # Robert Israel, Jun 12 2019

Accumulate@ DivisorSigma[1, Range[1, 109, 2]] (* Michael De Vlieger, Jun 09 2019 *)

terms(n) = my(s=0, i=0); for(k=0, n-1, if(i>=n, break); s+=sigma(2*k+1); print1(s, ", "); i++)
/* Print initial 50 terms as follows: */
terms(50) \\ Felix Fröhlich, Jun 08 2019

a(n) = sum(k=1, 2*n-1, if (k%2, sigma(k))); \\ Michel Marcus, Jun 08 2019

from math import isqrt
def A326123(n): return (-(s:=isqrt(r:=n<<1))**2*(s+1) + sum((q:=r//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) -(t:=isqrt(m:=n>>1))**2*(t+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))+3*((u:=isqrt(n))**2*(u+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,u+1))>>1) # Chai Wah Wu, Nov 01 2023

a(n) = A024916(2n) - A326124(n).

a(n) ~ Pi^2 * n^2 / 8. - Vaclav Kotesovec, Aug 18 2021

A239052 Sum of divisors of 4*n-2.

Original entry on oeis.org

3, 12, 18, 24, 39, 36, 42, 72, 54, 60, 96, 72, 93, 120, 90, 96, 144, 144, 114, 168, 126, 132, 234, 144, 171, 216, 162, 216, 240, 180, 186, 312, 252, 204, 288, 216, 222, 372, 288, 240, 363, 252, 324, 360, 270, 336, 384, 360, 294, 468, 306, 312, 576

Offset: 1

Omar E. Pol, Mar 09 2014

Bisection of A062731 (odd part).
a(n) is also the total number of cells in the n-th branch of the second quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-2). For the quadrants 1, 3, 4 see A112610, A239053, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270, see example.
We can find the spiral on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			Illustration of initial terms:
------------------------------------------------------
.        Branches of the spiral
.        in the second quadrant             n    a(n)
------------------------------------------------------
.
.                  _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|         4     24
.                 | |
.             12 _| |
.               |_ _|  _ _ _ _ _ _
.         12 _ _|     |  _ _ _ _ _|         3     18
.      _ _ _| |    9 _| |
.     |  _ _ _|  9 _|_ _|
.     | |      _ _| |      _ _ _ _
.     | |     |  _ _| 12 _|  _ _ _|         2     12
.     | |     | |      _|   |
.     | |     | |     |  _ _|
.     | |     | |     | |    3 _ _
.     | |     | |     | |     |  _|         1      3
.     |_|     |_|     |_|     |_|
.
For n = 4 the sum of divisors of 4*n-2 is 1 + 2 + 7 + 14 = A000203(14) = 24. On the other hand the parts of the symmetric representation of sigma(14) are [12, 12] and the sum of them is 12 + 12 = 24, equaling the sum of divisors of 14, so a(4) = 24.

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

Cf. A000203, A008438, A016825, A062731, A074400, A112610, A193553, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239050, A239053, A244050, A245092, A262626.

a[n_] := DivisorSigma[1, 4*n - 2]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

a(n) = A000203(4n-2) = A000203(A016825(n-1)).
a(n) = 3*A008438(n-1). - Joerg Arndt, Mar 09 2014
Sum_{k=1..n} a(k) = (3*Pi^2/8) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A239053 Sum of divisors of 4*n-1.

Original entry on oeis.org

4, 8, 12, 24, 20, 24, 40, 32, 48, 56, 44, 48, 72, 72, 60, 104, 68, 72, 124, 80, 84, 120, 112, 120, 156, 104, 108, 152, 144, 144, 168, 128, 132, 240, 140, 168, 228, 152, 192, 216, 164, 168, 260, 248, 180, 248, 216, 192, 336, 200, 240, 312, 212, 264, 296

Offset: 1

Omar E. Pol, Mar 09 2014

Bisection of A008438.
a(n) is also the total number of cells in the n-th branch of the third quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-1), see example. For the quadrants 1, 2, 4 see A112610, A239052, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

			Illustration of initial terms:
-----------------------------------------------------
.        Branches of the spiral
.        in the third quadrant             n    a(n)
-----------------------------------------------------
.     _       _       _       _
.    | |     | |     | |     | |
.    | |     | |     | |     |_|_ _
.    | |     | |     | |    2  |_ _|       1      4
.    | |     | |     |_|_     2
.    | |     | |    4    |_
.    | |     |_|_ _        |_ _ _ _
.    | |    6      |_      |_ _ _ _|       2      8
.    |_|_ _ _        |_   4
.   8      | |_ _      |
.          |_    |     |_ _ _ _ _ _
.            |_  |_    |_ _ _ _ _ _|       3     12
.           8  |_ _|  6
.                  |
.                  |_ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _|       4     24
.                 8
.
For n = 4 the sum of divisors of 4*n-1 is 1 + 3 + 5 + 15 = A000203(15) = 24. On the other hand the parts of the symmetric representation of sigma(15) are [8, 8, 8] and the sum of them is 8 + 8 + 8 = 24, equaling the sum of divisors of 15, so a(4) = 24.

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

Cf. A000203, A004767, A008438, A062731, A074400, A112610, A193553, A196020, A235791, A236104, A237270, A237591, A237593, A239050, A239052, A244050, A245092, A262626.

[SumOfDivisors(4*n-1): n in [1..60]]; // Vincenzo Librandi, Dec 07 2016

A239053:=n->numtheory[sigma](4*n-1): seq(A239053(n), n=1..80); # Wesley Ivan Hurt, Dec 06 2016

DivisorSigma[1,4*Range[60]-1] (* Harvey P. Dale, Dec 06 2016 *)
Table[DivisorSigma[1, 4 n - 1], {n, 100}] (* Vincenzo Librandi, Dec 07 2016 *)

a(n) = sigma(4*n-1); \\ Michel Marcus, Dec 07 2016

a(n) = A000203(4n-1) = A000203(A004767(n-1)).
a(n) = 4*A097723(n-1). - Joerg Arndt, Mar 09 2014
Sum_{k=1..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A247954 a(n) = sigma(sigma(2n-1)).

Original entry on oeis.org

1, 7, 12, 15, 14, 28, 24, 60, 39, 42, 63, 60, 32, 90, 72, 63, 124, 124, 60, 120, 96, 84, 168, 124, 80, 195, 120, 195, 186, 168, 96, 210, 224, 126, 252, 195, 114, 224, 252, 186, 133, 224, 280, 360, 234, 248, 255, 360, 171, 392, 216, 210, 508, 280, 216, 300

Offset: 1

Jaroslav Krizek, Sep 28 2014

See A247821 - numbers k such that sigma(sigma(2k-1)) is a prime p.

			For n=2; a(2) = sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000203, A051027, A008438, A247820.

[SumOfDivisors(SumOfDivisors(2*n-1)): n in [1..1000]];

with(numtheory): A247954:=n->sigma(sigma(2*n-1)): seq(A247954(n), n=1..50); # Wesley Ivan Hurt, Oct 01 2014

Table[DivisorSigma[1, DivisorSigma[1, 2 n - 1]], {n, 50}] (* Wesley Ivan Hurt, Oct 01 2014 *)

vector(100,n,sigma(sigma(2*n-1))) \\ Derek Orr, Sep 29 2014

a(n) = A000203(A000203(2n-1)) = A000203(A008438(n-1)) = A051027(2n-1).

cp's OEIS Frontend

A226255 Number of ways of writing n as the sum of 11 triangular numbers.

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A045823 a(n) = sigma_3(2*n+1).

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A109506 Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.

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A247820 Numbers k such that sigma(2*k-1) is a prime p.

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A340949 Number of ways to write n as an ordered sum of 4 nonzero triangular numbers.

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A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

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A326123 a(n) is the sum of all divisors of the first n odd numbers.

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