cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A351307 Sum of the squares of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 17, 1, 1, 1, 273, 1, 82, 1, 17, 1, 1, 1, 17, 626, 1, 82, 17, 1, 1, 1, 273, 1, 1, 1, 1394, 1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 273, 2402, 626, 1, 17, 1, 82, 1, 17, 1, 1, 1, 17, 1, 1, 82, 4369, 1, 1, 1, 17, 1, 1, 1, 1394, 1, 1, 626, 17, 1, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^2 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Examples

			a(16) = 273; a(16) = Sum_{d^2|16} (d^2)^2 = (1^2)^2 + (2^2)^2 + (4^2)^2 = 273.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), this sequence (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A010052, A247041 (zeta(5/2)), A008836, A001157.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*(1 + Floor[e/2])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
  • PARI
    my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022
  • Python
    from math import prod
from sympy import factorint
def A351307(n): return prod((p**(4+((e&-2)<<1))-1)//(p**4-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 11 2024

Formula

a(n) = Sum_{d^2|n} (d^2)^2.
Multiplicative with a(p) = (p^(4*(1+floor(e/2))) - 1)/(p^4 - 1). - Amiram Eldar, Feb 07 2022
G.f.: Sum_{k>0} k^4*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022
From Amiram Eldar, Sep 19 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-4).
Sum_{k=1..n} a(k) ~ (zeta(5/2)/5) * n^(5/2). (End)
a(n) = Sum_{d|n} d^2 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d^2*sigma_2(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A351308 Sum of the cubes of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 65, 1, 1, 1, 65, 730, 1, 1, 65, 1, 1, 1, 4161, 1, 730, 1, 65, 1, 1, 1, 65, 15626, 1, 730, 65, 1, 1, 1, 4161, 1, 1, 1, 47450, 1, 1, 1, 65, 1, 1, 1, 65, 730, 1, 1, 4161, 117650, 15626, 1, 65, 1, 730, 1, 65, 1, 1, 1, 65, 1, 1, 730, 266305, 1, 1, 1, 65, 1, 1, 1, 47450, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^3 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Examples

			a(16) = 4161; a(16) = Sum_{d^2|16} (d^2)^3 = (1^2)^3 + (2^2)^3 + (4^2)^3 = 4161.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), this sequence (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A010052, A261804 (zeta(7/2)), A008836, A001158.

Programs

  • Mathematica
    f[p_, e_] := (p^(6*(1 + Floor[e/2])) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
  • PARI
    a(n) = sumdiv(n, d, if (issquare(d), d^3)); \\ Michel Marcus, Mar 24 2023

Formula

a(n) = Sum_{d^2|n} (d^2)^3.
Multiplicative with a(p) = (p^(6*(1+floor(e/2))) - 1)/(p^6 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 19 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-6).
Sum_{k=1..n} a(k) ~ (zeta(7/2)/7) * n^(7/2). (End)
G.f.: Sum_{k>=1} k^6 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024
a(n) = Sum_{d|n} d^3 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d^3*sigma_3(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A358346 a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 9, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 36, 1, 42, 28, 8, 30, 72, 32, 33, 48, 54, 48, 1, 38, 60, 56, 54, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 84, 72, 72, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144
Offset: 1

Views

Author

Amiram Eldar, Nov 11 2022

Keywords

Comments

The number of unitary divisors of n that are exponentially odd is A055076(n).

Links

Crossrefs

Cf. A000290, A005117, A055076, A077610, A268335, A351569.
Similar sequences: A033634, A034448, A035316, A358347.

Programs

  • Mathematica
    f[p_, e_] := 1 + If[OddQ[e], p^e, 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i,2]%2,  f[i,1]^f[i,2], 0));}

Formula

a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A033634(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^e + 1 if e is odd, and 1 otherwise.
a(n) = A034448(n)/A358347(n).
Sum_{k=1..n} a(k) ~ n^2/2.
From Amiram Eldar, Sep 14 2023: (Start)
a(n) = A034448(A350389(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(2*s-1)). (End)

A351309 Sum of the 4th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 257, 1, 1, 1, 257, 6562, 1, 1, 257, 1, 1, 1, 65793, 1, 6562, 1, 257, 1, 1, 1, 257, 390626, 1, 6562, 257, 1, 1, 1, 65793, 1, 1, 1, 1686434, 1, 1, 1, 257, 1, 1, 1, 257, 6562, 1, 1, 65793, 5764802, 390626, 1, 257, 1, 6562, 1, 257, 1, 1, 1, 257, 1, 1, 6562, 16843009, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^4 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 29 2024

Examples

			a(16) = 65793; a(16) = Sum_{d^2|16} (d^2)^4 = (1^2)^4 + (2^2)^4 + (4^2)^4 = 65793.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), this sequence (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A010052, A008836, A001159.

Programs

  • Mathematica
    f[p_, e_] := (p^(8*(1 + Floor[e/2])) - 1)/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^4],{n,70}] (* Harvey P. Dale, Feb 11 2023 *)
  • PARI
    a(n) = sumdiv(n, d, if (issquare(d), d^4)); \\ Michel Marcus, Jun 05 2024

Formula

a(n) = Sum_{d^2|n} (d^2)^4.
Multiplicative with a(p) = (p^(8*(1+floor(e/2))) - 1)/(p^8 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-8).
Sum_{k=1..n} a(k) ~ (zeta(9/2)/9) * n^(9/2). (End)
G.f.: Sum_{k>=1} k^8 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024
a(n) = Sum_{d|n} d^4 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 29 2024
a(n) = Sum_{d|n} lambda(d)*d^4*sigma_4(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351310 Sum of the 5th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1025, 1, 1, 1, 1049601, 1, 59050, 1, 1025, 1, 1, 1, 1025, 9765626, 1, 59050, 1025, 1, 1, 1, 1049601, 1, 1, 1, 60526250, 1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1049601, 282475250, 9765626, 1, 1025, 1, 59050, 1, 1025, 1, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^5 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(16) = 1049601; a(16) = Sum_{d^2|16} (d^2)^5 = (1^2)^5 + (2^2)^5 + (4^2)^5 = 1049601.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), this sequence (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A010052, A008836, A001160.

Programs

  • Mathematica
    f[p_, e_] := (p^(10*(1 + Floor[e/2])) - 1)/(p^10 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

Formula

a(n) = Sum_{d^2|n} (d^2)^5.
Multiplicative with a(p) = (p^(10*(1+floor(e/2))) - 1)/(p^10 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-10).
Sum_{k=1..n} a(k) ~ (zeta(11/2)/11) * n^(11/2). (End)
a(n) = Sum_{d|n} d^5 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^5*sigma_5(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351311 Sum of the 6th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 4097, 1, 1, 1, 16781313, 1, 531442, 1, 4097, 1, 1, 1, 4097, 244140626, 1, 531442, 4097, 1, 1, 1, 16781313, 1, 1, 1, 2177317874, 1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 16781313, 13841287202, 244140626, 1, 4097, 1, 531442, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^6 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(16) = 16781313; a(16) = Sum_{d^2|16} (d^2)^6 = (1^2)^6 + (2^2)^6 + (4^2)^6 = 16781313.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), this sequence (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A010052, A008836, A013954.

Programs

  • Mathematica
    f[p_, e_] := (p^(12*(1 + Floor[e/2])) - 1)/(p^12 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

Formula

a(n) = Sum_{d^2|n} (d^2)^6.
Multiplicative with a(p) = (p^(12*(1+floor(e/2))) - 1)/(p^12 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-12).
Sum_{k=1..n} a(k) ~ (zeta(13/2)/13) * n^(13/2). (End)
a(n) = Sum_{d|n} d^6 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^6*sigma_6(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351313 Sum of the 7th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 16385, 1, 1, 1, 16385, 4782970, 1, 1, 16385, 1, 1, 1, 268451841, 1, 4782970, 1, 16385, 1, 1, 1, 16385, 6103515626, 1, 4782970, 16385, 1, 1, 1, 268451841, 1, 1, 1, 78368963450, 1, 1, 1, 16385, 1, 1, 1, 16385, 4782970, 1, 1, 268451841, 678223072850, 6103515626, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^7 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(16) = 268451841; a(16) = Sum_{d^2|16} (d^2)^7 = (1^2)^7 + (2^2)^7 + (4^2)^7 = 268451841.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), this sequence (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A010052, A008836, A013955.

Programs

  • Mathematica
    f[p_, e_] := (p^(14*(1 + Floor[e/2])) - 1)/(p^14 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

Formula

a(n) = Sum_{d^2|n} (d^2)^7.
Multiplicative with a(p) = (p^(14*(1+floor(e/2))) - 1)/(p^14 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-14).
Sum_{k=1..n} a(k) ~ (zeta(15/2)/15) * n^(15/2). (End)
a(n) = Sum_{d|n} d^7 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^7*sigma_7(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351314 Sum of the 8th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 65537, 1, 1, 1, 4295032833, 1, 43046722, 1, 65537, 1, 1, 1, 65537, 152587890626, 1, 43046722, 65537, 1, 1, 1, 4295032833, 1, 1, 1, 2821153019714, 1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 4295032833, 33232930569602, 152587890626
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^8 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(16) = 4295032833; a(16) = Sum_{d^2|16} (d^2)^8 = (1^2)^8 + (2^2)^8 + (4^2)^8 = 4295032833.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), this sequence (k=8), A351315 (k=9), A351316 (k=10).
Cf. A010052, A008836, A013956.

Programs

  • Mathematica
    f[p_, e_] := (p^(16*(1 + Floor[e/2])) - 1)/(p^16 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^8],{n,80}] (* Harvey P. Dale, Feb 13 2022 *)
  • PARI
    my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^16*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022

Formula

a(n) = Sum_{d^2|n} (d^2)^8.
Multiplicative with a(p) = (p^(16*(1+floor(e/2))) - 1)/(p^16 - 1). - Amiram Eldar, Feb 07 2022
G.f.: Sum_{k>0} k^16*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-16).
Sum_{k=1..n} a(k) ~ (zeta(17/2)/17) * n^(17/2). (End)
a(n) = Sum_{d|n} d^8 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^8*sigma_8(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351315 Sum of the 9th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 262145, 1, 1, 1, 262145, 387420490, 1, 1, 262145, 1, 1, 1, 68719738881, 1, 387420490, 1, 262145, 1, 1, 1, 262145, 3814697265626, 1, 387420490, 262145, 1, 1, 1, 68719738881, 1, 1, 1, 101560344351050, 1, 1, 1, 262145, 1, 1, 1, 262145, 387420490, 1, 1, 68719738881
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

Comments

Inverse Möbius transform of n^9 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(16) = 68719738881; a(16) = Sum_{d^2|16} (d^2)^9 = (1^2)^9 + (2^2)^9 + (4^2)^9 = 68719738881.

Links

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), this sequence (k=9), A351316 (k=10).
Cf. A010052, A008836, A013957.

Programs

  • Mathematica
    f[p_, e_] := (p^(18*(1 + Floor[e/2])) - 1)/(p^18 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
snp[n_]:=Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^9]; Array[snp,50] (* Harvey P. Dale, May 25 2025 *)

Formula

a(n) = Sum_{d^2|n} (d^2)^9.
Multiplicative with a(p) = (p^(18*(1+floor(e/2))) - 1)/(p^18 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-18).
Sum_{k=1..n} a(k) ~ (zeta(19/2)/19) * n^(19/2). (End)
a(n) = Sum_{d|n} d^9 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^9*sigma_9(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A185027 Sum of the triangular divisors of n.

Original entry on oeis.org

1, 1, 4, 1, 1, 10, 1, 1, 4, 11, 1, 10, 1, 1, 19, 1, 1, 10, 1, 11, 25, 1, 1, 10, 1, 1, 4, 29, 1, 35, 1, 1, 4, 1, 1, 46, 1, 1, 4, 11, 1, 31, 1, 1, 64, 1, 1, 10, 1, 11, 4, 1, 1, 10, 56, 29, 4, 1, 1, 35, 1, 1, 25, 1, 1, 76, 1, 1, 4, 11, 1, 46, 1, 1, 19, 1, 1, 88, 1
Offset: 1

Views

Author

Antonio Roldán, Jan 14 2013

Keywords

Examples

			a(15) = 19 because 1+3+15 = 19 (1, 3 and 15 are the triangular divisors of 15).

Links

Crossrefs

Cf. A000217, A035316.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, IntegerQ[Sqrt[8*#+1]] &]; Array[a, 100] (* Amiram Eldar, Aug 12 2023 *)
  • PARI
    istriang(x)=issquare(8*x+1)
a(n)={my(m=0);for(i=1,n,if(istriang(i)&&n/i==n\i,m+=i));return(m)}
{for(n=1,10^4,k=sumdivtriang(n);write("b185027.txt",n," ",k))}
  • PARI
    a(n)=sumdiv(n, d, ispolygonal(d, 3)*d) \\ Charles R Greathouse IV, Jan 14 2013

Formula

G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^(k*(k+1)/2)/(1 - x^(k*(k+1)/2)). - Ilya Gutkovskiy, Dec 24 2016
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