A001082 -id:A001082 - cp's OEIS frontend

A308662 Number of ways to write n as (2^a5^b)^2 + c(3c+1) + d*(3d+2), where a and b are nonnegative integers, and c and d are integers.

1, 1, 1, 2, 2, 3, 1, 2, 3, 1, 3, 2, 2, 2, 2, 4, 2, 2, 5, 3, 3, 3, 3, 3, 4, 6, 4, 3, 3, 5, 4, 4, 3, 6, 5, 6, 3, 2, 6, 3, 6, 2, 3, 4, 4, 6, 5, 5, 4, 4, 6, 1, 4, 4, 4, 6, 3, 5, 2, 6, 7, 3, 2, 5, 5, 4, 5, 6, 8, 5, 6, 5, 4, 8, 3, 7, 3, 3, 7, 3, 6, 7, 4, 4, 7, 7, 4, 4, 8, 7, 4, 3, 6, 4, 7, 7, 4, 1, 6, 7

Offset: 1

Zhi-Wei Sun, Jun 15 2019

nonn

Conjecture 1: a(n) > 0 for all n > 0.
Conjecture 2: Let r be 1 or 2. Then, any positive integer n can be written as (2^a*5^b)^2 + c*(2c+1) + d*(3d+r), where a and b are nonnegative integers, and c and d are integers.
We have verified Conjectures 1 and 2 for all n = 1..10^8.

			a(3) = 1 with 3 = (2^0*5^0)^2 + (-1)*(3*(-1)+1) + 0*(3*0+2).
a(7) = 1 with 7 = (2^1*5^0)^2 + (-1)*(3*(-1)+1) + (-1)*(3*(-1)+2).
a(10) = 1 with 10 = (2^0*5^0)^2 + 1*(3*1+1) + 1*(3*1+2).
a(52) = 1 with 52 = (2^0*5^0)^2 + 3*(3*3+1) + (-3)*(3*(-3)+2).
a(98) = 1 with 98 = (2^0*5^1)^2 + 4*(3*4+1) + (-3)*(3*(-3)+2).
a(14596) = 1 with 14596 = (2^3*5^0)^2 + (-36)*(3*(-36)+1) + (-60)*(3*(-60)+2).
a(22163) = 1 with 22163 = (2^3*5^0)^2 + 66*(3*66+1) + (-55)*(3*(-55)+2).
a(150689) = 1 with 150689 = (2^6*5^1)^2 + 117*(3*117+1) + (-49)*(3*(-49)+2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58 (2015), No. 7, 1367-1396.

Cf. A000079, A000351, A001082, A001318, A308566, A308584, A308621, A308623, A308640, A308641, A308644, A308656, A308661.

OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]];
tab={};Do[r=0;Do[If[OctQ[n-4^a*25^b-x(3x+1)],r=r+1],{a,0,Log[4,n]},{b,0,Log[25,n/4^a]},{x,-Floor[(Sqrt[12(n-4^a*25^b)+1]+1)/6],(Sqrt[12(n-4^a*25^b)+1]-1)/6}];tab=Append[tab,r],{n,1,100}];Print[tab]

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

Original entry on oeis.org

0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 1 we have A317301.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Row 0 of A303301.
Cf. A001057, A008795, A008794, A174474, A317301.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018

a(n) = -A174474(n+1).
From Colin Barker, Aug 01 2018: (Start)
G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = -n*(n + 4) / 4 for n even.
a(n) = -(n - 3)*(n + 1) / 4 for n odd.
(End)

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

Original entry on oeis.org

0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20, -9, -27, -14, -35, -20, -44, -27, -54, -35, -65, -44, -77, -54, -90, -65, -104, -77, -119, -90, -135, -104, -152, -119, -170, -135, -189, -152, -209, -170, -230, -189, -252, -209, -275, -230, -299, -252, -324, -275, -350, -299, -377, -324, -405, -350, -434

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 0 we have A317300.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Row 1 of A303301.
Cf. A000096, A001057, A008794, A008795, A317300.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

/* By definition: */ k:=1; [0] cat [m*i*((k-2)*m*i-k+4)/2: i in [1,-1], m in [1..30]]; // Bruno Berselli, Jul 30 2018

Table[(-2 n (n + 1) - 5 (2 n + 1) (-1)^n + 5)/16, {n, 0, 60}] (* Bruno Berselli, Jul 30 2018 *)

concat(0, Vec(x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^50))) \\ Colin Barker, Aug 01 2018

From Bruno Berselli, Jul 30 2018: (Start)
O.g.f.: x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (-5*(1 + 2*x) + (5 - 2*x^2)*exp(2*x))*exp(-x)/16.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (-2*n*(n + 1) - 5*(2*n + 1)*(-1)^n + 5)/16. Therefore:
a(n) = -n*(n + 6)/8 for even n;
a(n) = -(n - 5)*(n + 1)/8 for odd n. Also:
a(n) = a(n-5) for odd n > 3.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) + n*(n^2 - 3) = 0. (End)

A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 6, 8, 10, 10, 12, 16, 22, 28, 32, 36, 42, 52, 66, 80, 92, 104, 120, 144, 174, 206, 236, 266, 304, 356, 420, 488, 554, 624, 708, 816, 946, 1084, 1224, 1372, 1548, 1764, 2016, 2288, 2568, 2868, 3216, 3632, 4110, 4626, 5166, 5748, 6412, 7188

Offset: 0

Noureddine Chair, Feb 15 2005

Convolution of A098884 and A003105. [corrected by Vaclav Kotesovec, Feb 07 2021]

Also equal to the number of overpartitions of n into parts congruent to 1 or 5 modulo 6. - Jeremy Lovejoy, Nov 28 2024

			E.g. a(7)=8 because 14=10+4=10+2+1+1=8+4+2=8+4+1+1=7+7=5+5+4=5+5+2+1+1.

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A003105, A098884, A080054, A001082, A098151, A015128.

series(product(((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))),k=1..100),x=0,100);
# alternative program:
with(gfun): series( add(x^(n*(3*n-2)), n = -6..6)/add((-1)^n*x^(n*(3*n-2)), n = -6..6), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

nmax = 50; CoefficientList[Series[Product[((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

G.f.: (Theta_4(0, x^2)*theta_4(0, x^3))/(theta_4(0, x)*theta_4(0, x^(6))) = Product_{k>0}((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))).
Euler transform of period 12 sequence [2, -1, 0, 0, 2, 0, 2, 0, 0, -1, 2, 0, ...]. - Vladeta Jovovic, Feb 17 2005
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 01 2015
G.f.: f(x,x^5)/f(-x,-x^5) = ( Sum_{n = -oo..oo} x^(n*(3*n-2)) )/( Sum_{n = -oo..oo} (-1)^n*x^(n*(3*n-2)) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. Cf. A080054 and A098151. - Peter Bala, Feb 05 2021

Example corrected by Vaclav Kotesovec, Sep 01 2015

A104566 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product H*R of the infinite lower triangular matrices H = [1; 1,2; 1,2,1; 1 2,1,2; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 15 2005

			The first few rows are
  1;
  3, 2;
  4, 3, 1;
  6, 5, 3, 2;
  ...

Row sums yield A001082.
Columns 1, 3, 5, ... (starting at the diagonal entry) yield A032766.
Columns 2, 4, 6, ... (starting at the diagonal entry) yield A045506.
Row sums = 1, 5, 8, 16, 21, ... (generalized octagonal numbers, A001082). A006578(2n-1) = A001082(2n).

T:=proc(i,j) if j>i then 0 elif i mod 2 = 1 and j mod 2 = 1 then 3*(i-j)/2+1 elif i mod 2 = 0 and j mod 2 = 0 then 3*(i-j)/2+2 elif i+j mod 2 = 1 then 3*(i-j+1)/2 else fi end: for i from 1 to 14 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 24 2005

For 1 <= j <= i: T(i,j) = 3(i-j+1)/2 if i and j are of opposite parity; T(i,j) = 3(i-j)/2 + 1 if both i and j are odd; T(i,j) = 3(i-j)/2 + 2 if both i and j are even. - Emeric Deutsch, Mar 24 2005

More terms from Emeric Deutsch, Mar 24 2005

A132354 Integers m such that 7*m + 1 is a square.

Original entry on oeis.org

0, 5, 9, 24, 32, 57, 69, 104, 120, 165, 185, 240, 264, 329, 357, 432, 464, 549, 585, 680, 720, 825, 869, 984, 1032, 1157, 1209, 1344, 1400, 1545, 1605, 1760, 1824, 1989, 2057, 2232, 2304, 2489, 2565, 2760, 2840, 3045, 3129, 3344, 3432, 3657, 3749, 3984, 4080

Offset: 0

Mohamed Bouhamida, Nov 08 2007

Numbers of the form m*(7*m + 2) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A054000, A056220, A000217, A087475, A028347, A062717, A036666, A002378, A001082, A046092, A005563.

[n^2+n+3*Ceiling(n/2)^2: n in [0..50]]; // Wesley Ivan Hurt, Jul 07 2014

seq(n^2+n+3*ceil(n/2)^2, n=0..48); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+7*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

a(2*k) = k*(7*k + 2), a(2*k + 1) = 7*k^2 + 12*k + 5.
a(n) = n^2 + n + 3*ceiling(n/2)^2. - Gary Detlefs, Feb 23 2010
G.f.: -x*(5*x^2 + 4*x + 5)/((x - 1)^3*(x + 1)^2). - Colin Barker, Oct 24 2012
Sum_{n>=1} 1/a(n) = 7/4 - cot(2*Pi/7)*Pi/2. - Amiram Eldar, Mar 15 2022

More terms from Max Alekseyev, Nov 13 2009

Better definition from Max Alekseyev, Oct 24 2012

A185124 Expansion of f(x, -x^5) in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jan 20 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is a number of A001082.
The exponents in the q-series for this sequence are the squares of the numbers of A001651.

			G.f. = 1 + x - x^5 - x^8 - x^16 - x^21 + x^33 + x^40 + x^56 + x^65 - x^85 + ...
G.f. = q + q^4 - q^16 - q^25 - q^49 - q^64 + q^100 + q^121 + q^169 + q^196 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A010815, A080902, A185125.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^6] QPochhammer[ x^5, -x^6] QPochhammer[ -x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)

{a(n) = my(m); if( issquare( 3*n + 1, &m), (m%3!=0) * (-1)^((m+3) \ 6), 0)};

Euler transform of period 24 sequence [ 1, -1, 0, 0, -1, 1, -1, 0, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, -1, ...].
G.f.: Sum_{k in Z} (-1)^floor((k + 1)/2) * x^(k * (3*k + 2)).
a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(4*n + 1) = A080902(n). a(8*n) = A010815(n).
a(n) = (-1)^n * A185125(n). - Michael Somos, Jun 30 2015

A185125 Expansion of f(-x, x^5) in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jan 20 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is a number of A001082.
The exponents in the q-series for this sequence are the squares of the numbers of A001651.

			G.f. = 1 - x + x^5 - x^8 - x^16 + x^21 - x^33 + x^40 + x^56 - x^65 + x^85 + ...
G.f. = q - q^4 + q^16 - q^25 - q^49 + q^64 - q^100 + q^121 + q^169 - q^196 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A010815, A080902, A185124.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x^6] QPochhammer[ -x^5, -x^6] QPochhammer[ -x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)

{a(n) = my(m); if( issquare( 3*n + 1, &m), (m%3!=0) * (-1)^((m+3) \ 6 + n), 0)};

Euler transform of period 24 sequence [ -1, 0, 0, 0, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 0, 0, 0, -1, -1, ...].
G.f.: Sum_{k in Z} (-1)^floor(k/2) * x^(k * (3*k + 2)).
a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(4*n + 1) = - A080902(n). a(8*n) = A010815(n).
a(n) = (-1)^n * A185124(n). - Michael Somos, Jun 30 2015

A185918 a(n) = 12n^2 - 2n - 1.

Original entry on oeis.org

-1, 9, 43, 101, 183, 289, 419, 573, 751, 953, 1179, 1429, 1703, 2001, 2323, 2669, 3039, 3433, 3851, 4293, 4759, 5249, 5763, 6301, 6863, 7449, 8059, 8693, 9351, 10033, 10739, 11469, 12223, 13001, 13803, 14629, 15479, 16353, 17251, 18173, 19119, 20089, 21083, 22101, 23143, 24209

Offset: 0

Paul Curtz, Feb 08 2011

The second quadrisection of A184005(n-1) is A179741(n).
The first quadrisection of A184005(n-1) is a(n).
Sequence found by reading the line from -1, in the direction -1, 9, ..., in the square spiral whose vertices are -1 together with the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001082, A154106, A179741, A184005.

[-1-2*n+12*n^2: n in [0..80] ]; // Vincenzo Librandi, Feb 09 2011

A185918:=n->12*n^2-2*n-1: seq(A185918(n), n=0..60); # Wesley Ivan Hurt, Jan 31 2017

Table[12n^2-2n-1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{-1,9,43},50] (* Harvey P. Dale, May 20 2012 *)

a(n)=12*n^2-2*n-1 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = A184005(4*n-1). [corrected by R. J. Mathar, Aug 24 2011]
a(n) = a(n-1) + 24*n - 14.
a(n) = 2*a(n-1) - a(n) + 24.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(1+x)*(13*x-1) / (x-1)^3. - R. J. Mathar, Aug 24 2011
a(n) = A154106(n-1) - 2, n >= 1. - Omar E. Pol, Jul 19 2012
E.g.f.: (12*x^2 + 10*x -1)*exp(x). - G. C. Greubel, Jul 22 2017

More terms from Vincenzo Librandi, Feb 09 2011

A245028 Divisors of 11^12 - 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 26, 28, 30, 35, 36, 37, 38, 39, 40, 42, 45, 48, 52, 56, 57, 60, 61, 63, 65, 70, 72, 74, 76, 78, 80, 84, 90, 91, 95, 104, 105, 111, 112, 114, 117, 120, 122, 126, 130, 133, 140, 144, 148, 152

Offset: 1

Bruno Berselli, Jul 10 2014

See Comments section in A245027.
The following 36 triangular numbers belong to this sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, 78, 91, 105, 120, 171, 190, 210, 630, 666, 703, 741, 780, 1596, 1830, 4095, 4560, 5460, 6216, 16653, 33670, 46360, 103740, 115440, 221445, 274170, 365085, 392303547090.
The following terms of A001082 (without 1, 21 and 120, which appear in the previous list) are in sequence: 5, 8, 16, 40, 56, 65, 133, 208, 280, 456, 481, 560, 936, 1008, 1281, 1365, 1680, 1776, 1976, 4880, 5985, 10920, 11285, 44408, 47880, 590520, 658008, 731120, 973560, 1046142792240.
Also, 4/5 of the members are divisible by 3 and 2/3 of them are even.

			3138428376720 = 2^4 * 3^2 * 5 * 7 * 13 * 19 * 37 * 61 * 1117.

Bruno Berselli, Table of n, a(n) for n = 1..1920
Index entries for sequences related to divisors of numbers

Cf. Divisors of k^12-1: A003524 (k=2); A003532 (k=4); A245027 (k=7), A003543 (k=8), A027902 (k=9), A027897 (k=10).

Magma
```
Divisors(11^12-1);
```
Mathematica
```
Divisors[11^12 - 1]
```
Maxima
```
divisors(11^12-1);
```
PARI
```
divisors(11^12-1)
```
Sage
```
divisors(11^12-1)
```

cp's OEIS Frontend

A308662 Number of ways to write n as (2^a*5^b)^2 + c*(3c+1) + d*(3d+2), where a and b are nonnegative integers, and c and d are integers.

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A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

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A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

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A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

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A104566 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product H*R of the infinite lower triangular matrices H = [1; 1,2; 1,2,1; 1 2,1,2; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

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A132354 Integers m such that 7*m + 1 is a square.

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A185124 Expansion of f(x, -x^5) in powers of x where f(,) is the Ramanujan general theta function.

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A308662 Number of ways to write n as (2^a5^b)^2 + c(3c+1) + d*(3d+2), where a and b are nonnegative integers, and c and d are integers.

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

A185918 a(n) = 12n^2 - 2n - 1.