A185027 -id:A185027 - cp's OEIS frontend

A209309 Numbers whose sum of triangular divisors is also triangular and greater than 1.

6, 12, 18, 24, 48, 54, 96, 102, 110, 114, 138, 162, 174, 186, 192, 204, 220, 222, 228, 246, 258, 282, 315, 318, 348, 354, 364, 366, 372, 384, 402, 414, 426, 438, 440, 444, 456, 474, 486, 492, 498, 516, 522, 534, 550, 558, 564, 582, 606, 618, 636, 642, 654, 678

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			186 is a term because the sum of its triangular divisors, 1+3+6 = 10 is also triangular.

Antonio Roldán, Table of n, a(n) for n = 1..8055 (terms <= 10^5)

Cf. A000217, A035316, A185027, A209310.

triQ[n_] := n > 1 && IntegerQ[Sqrt[8*n+1]]; q[n_] := triQ[1 + DivisorSum[n, #&, triQ[#] &]]; Select[Range[700], q] (* Amiram Eldar, Aug 12 2023 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^5, k=sumdiv(n, d, istriangular(d)*d); if(istriangular(k)&&k>>1, t+=1; write("b209309.txt",t," ",n)))}

A209310 Triangular numbers whose sum of triangular divisors is also triangular and greater than 1.

Original entry on oeis.org

6, 4186, 32131, 52975, 78210, 111628, 237016, 247456, 584821, 750925, 1464616, 3649051, 5791906, 11297881, 16082956, 24650731, 27243271, 38618866, 46585378, 51546781, 56026405, 76923406, 89880528, 96070591, 126906346, 164629585, 201854278, 228733966

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			4186 is in sequence because it is triangular (4186 = 91*92/2) and the sum of its triangular divisors, 4186+91+1 = 4278 is also triangular (4278 = 92*93/2).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Subsequence of A209309.

Cf. A000217, A185027, A035316.

triQ[n_] := n > 1 && IntegerQ[Sqrt[8*n+1]]; q[n_] := triQ[1 + DivisorSum[n, #&, triQ[#] &]]; Select[Accumulate[Range[22000]], q] (* Amiram Eldar, Aug 12 2023 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^8, if(istriangular(n), k=sumdiv(n, d, istriangular(d)*d) ;if(istriangular(k)&&k>>1,t+=1;write("b209310.txt",t," ",n))))}

A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k(k+1)/2))) = Sum_{n>=1} a(n)x^n/n.

Original entry on oeis.org

1, -1, 4, -1, 1, 2, 1, -1, 4, 9, 1, -10, 1, -1, 19, -1, 1, 2, 1, -11, 25, -1, 1, -10, 1, -1, 4, 27, 1, -3, 1, -1, 4, -1, 1, 26, 1, -1, 4, -11, 1, -19, 1, -1, 64, -1, 1, -10, 1, 9, 4, -1, 1, 2, 56, -29, 4, -1, 1, -35, 1, -1, 25, -1, 1, 68, 1, -1, 4, 9, 1, -46, 1, -1, 19, -1, 1

Offset: 1

Ilya Gutkovskiy, May 20 2018

sign

			L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 - x^8/8 + 4*x^9/9 + 9*x^10/10 + ...
exp(L(x)) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2*x^10 + x^11 + x^13 + x^14 + x^15 + 2*x^16 + ... + A024940(n)*x^n + ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A000217, A010054, A024940, A185027, A300853.

nmax = 77; Rest[CoefficientList[Series[Log[Product[1 + x^(k (k + 1)/2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 77; Rest[CoefficientList[Series[Sum[k (k + 1)/2 x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(8 # + 1)^(1/2)] &], {n, 77}]

A010054(n) = issquare(8*n + 1);
A304876(n) = sumdiv(n,d,(-1)^(1+(n/d)) * A010054(d)*d); \\ Antti Karttunen, Feb 20 2023

G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^(k*(k+1)/2)/(1 + x^(k*(k+1)/2)).

a(n) = Sum_{d|n} (-1)^(n/d+1)*A010054(d)*d.

A327629 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 - x^(k(k + 1)/2))^2.

Original entry on oeis.org

1, 2, 4, 4, 5, 9, 7, 8, 12, 11, 11, 18, 13, 14, 21, 16, 17, 27, 19, 22, 29, 22, 23, 36, 25, 26, 36, 29, 29, 50, 31, 32, 44, 34, 35, 55, 37, 38, 52, 44, 41, 65, 43, 44, 64, 46, 47, 72, 49, 55, 68, 52, 53, 81, 56, 58, 76, 58, 59, 100, 61, 62, 87, 64, 65, 100, 67, 68, 92, 77

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

nonn

Sum of divisors d of n such that n/d is triangular number.

Cf. A000217, A006463, A007862, A010054, A076752, A112886 (fixed points), A185027.

nmax = 70; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Sqrt[8 n/# + 1]] &]; Table[a[n], {n, 1, 70}]

a(n)={sumdiv(n, d, if(ispolygonal(d,3), n/d))} \\ Andrew Howroyd, Sep 19 2019

from sympy import divisors
from sympy.ntheory.primetest import is_square
def A327629(n): return sum(n//d for d in divisors(n,generator=True) if is_square((d<<3)+1)) # Chai Wah Wu, Jun 07 2025

a(n) = Sum_{d|n} A010054(n/d) * d.

A209311 Numbers whose sum of triangular divisors is also a divisor and greater than 1.

Original entry on oeis.org

285, 1302, 1425, 1820, 2508, 3640, 3720, 4845, 4956, 5016, 5415, 7125, 7280, 9100, 9114, 9912, 11685, 12255, 12740, 14508, 15105, 16815, 17385, 18200, 19095, 19824, 20235, 20805, 22134, 22515, 23655, 23660, 24021, 24738, 25365, 25480, 27075, 27588, 27645

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			285 is in the sequence because its divisors being 1, 3, 5, 15, 19, 57, 95, 285, of which 1, 3 and 15 are triangular, these add up to 19.
1302 is in sequence because the sum of triangular divisors 21 + 6 + 3 + 1 = 31 is divisor of 1302.

Antonio Roldán, Table of n, a(n) for n = 1..13202 (terms < 10^7)

Cf. A185027, A035316, A209309, A209310.

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; fQ[n_] := Module[{tri = Total[Select[Divisors[n], TriangularQ]]}, tri > 1 && Mod[n, tri] == 0]; Select[Range[28000], fQ] (* T. D. Noe, Jan 23 2013 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^7, k=sumdiv(n, d, istriangular(d)*d); if(n/k==n\k&&k>>1, t+=1; write("b209311.txt",t," ",n)))}

A334987 Sum of centered triangular numbers dividing n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 20, 15, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 32, 5, 1, 1, 1, 5, 1, 20, 1, 15, 1, 1, 1, 5, 1, 47, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 20, 1, 1, 15, 1, 32, 1, 69, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 24, 1, 1, 1, 15, 1, 1, 1, 5, 86, 1, 1, 5, 1, 11

Offset: 1

Ilya Gutkovskiy, May 18 2020

nonn

Eric Weisstein's World of Mathematics, Centered Triangular Number

Cf. A005448, A185027, A280950, A300409, A334988.

nmax = 90; CoefficientList[Series[Sum[(3 k (k - 1)/2 + 1) x^(3 k (k - 1)/2 + 1)/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

isc(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k));
a(n) = sumdiv(n, d, if (isc(d), d)); \\ Michel Marcus, May 19 2020

G.f.: Sum_{k>=1} (3*k*(k - 1)/2 + 1) * x^(3*k*(k - 1)/2 + 1) / (1 - x^(3*k*(k - 1)/2 + 1)).

L.g.f.: log(G(x)), where G(x) is the g.f. for A280950.

A334988 Sum of tetrahedral numbers dividing n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 36, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 61, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 46, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 89, 1, 1, 1, 5, 1, 11

Offset: 1

Ilya Gutkovskiy, May 18 2020

nonn

Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A023533, A068980, A185027, A279495, A334987.

nmax = 90; CoefficientList[Series[Sum[Binomial[k + 2, 3] x^Binomial[k + 2, 3]/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

ist(n) = my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n; \\ A000292
a(n) = sumdiv(n, d, if (ist(d), d)); \\ Michel Marcus, May 19 2020

G.f.: Sum_{k>=1} binomial(k+2,3) * x^binomial(k+2,3) / (1 - x^binomial(k+2,3)).
L.g.f.: log(G(x)), where G(x) is the g.f. for A068980.
a(n) = Sum_{d|n} A023533(d) * d.

A343408 Sum of proper divisors of n that are triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 10, 1, 1, 4, 1, 1, 10, 1, 11, 4, 1, 1, 10, 1, 1, 4, 1, 1, 35, 1, 1, 4, 1, 1, 10, 1, 1, 4, 11, 1, 31, 1, 1, 19, 1, 1, 10, 1, 11, 4, 1, 1, 10, 1, 29, 4, 1, 1, 35, 1, 1, 25, 1, 1, 10, 1, 1, 4, 11, 1, 46, 1, 1, 19, 1, 1, 10, 1, 11, 4, 1, 1, 59, 1, 1, 4, 1, 1, 80, 1, 1, 4, 1, 1, 10

Offset: 1

Ilya Gutkovskiy, Apr 14 2021

nonn

Cf. A000217, A001065, A010054, A185027, A293235, A343407.

a:= n-> add(`if`(issqr(8*d+1), d, 0), d=numtheory[divisors](n) minus {n}):
seq(a(n), n = 1..96);  # Alois P. Heinz, Apr 14 2021

nmax = 96; CoefficientList[Series[Sum[(k (k + 1)/2) x^(k (k + 1))/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[If[d < n && IntegerQ[Sqrt[8 d + 1]], d, 0], {d, Divisors[n]}], {n, 96}]

a(n) = sumdiv(n, d, if ((dMichel Marcus, Apr 14 2021

G.f.: Sum_{k>=1} (k*(k+1)/2) * x^(k*(k+1)) / (1 - x^(k*(k+1)/2)).

a(n) = Sum_{d|n, d < n} A010054(d) * d.

cp's OEIS Frontend

A209309 Numbers whose sum of triangular divisors is also triangular and greater than 1.

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A209310 Triangular numbers whose sum of triangular divisors is also triangular and greater than 1.

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A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k*(k+1)/2))) = Sum_{n>=1} a(n)*x^n/n.

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A327629 Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 - x^(k*(k + 1)/2))^2.

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A209311 Numbers whose sum of triangular divisors is also a divisor and greater than 1.

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A334987 Sum of centered triangular numbers dividing n.

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A334988 Sum of tetrahedral numbers dividing n.

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A343408 Sum of proper divisors of n that are triangular numbers.

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A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k(k+1)/2))) = Sum_{n>=1} a(n)x^n/n.

A327629 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 - x^(k(k + 1)/2))^2.