A088915 -id:A088915 - cp's OEIS frontend

A121741 Dimensions of the irreducible representations of the simple Lie algebra of type A2 (equivalently, the group SL3) over the complex numbers, listed in increasing order.

1, 3, 6, 8, 10, 15, 21, 24, 27, 28, 35, 36, 42, 45, 48, 55, 60, 63, 64, 66, 78, 80, 81, 90, 91, 99, 105, 120, 125, 132, 136, 143, 153, 154, 162, 165, 168, 171, 190, 192, 195, 210, 216, 224, 231, 234, 253, 255, 260, 270, 273, 276, 280, 288, 300

Offset: 1

Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006, Aug 23 2006

nonn

We include "1" for the 1-dimensional trivial representation and we list each dimension once, ignoring the fact that inequivalent representations may have the same dimension.
Numbers of the form (x * (x - y) * (x - z) + y * (y - x) * (y - z) + z * (z - x) * (z - y)) / 18 with x + y + z = 0 and x * y * z > 0. - Michael Somos, Jun 26 2013
Positive numbers of the form (r-s)*r*(r+s) where r and s are integers, i.e., the product of three integers in arithmetic progression. In the expression above, set x = r-s, y = r+s, and z = -x-y. - Elliott Line, Dec 22 2020

N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.
J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Java Program
Wikipedia, The special linear group

Equals A088915(n+1)/2.

Cf. A121732, A121736, A121737, A121738, A121739, A104599, A121214, A162651.

GAP
```
# see program at sequence A121732
```

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A121741_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        for k in divisors(m<<1,generator=True):
            p, q = integer_nthroot(k**4+(k*m<<3),2)
            if q and not (p-k**2)%(k<<1):
                yield m
                break
A121741_list = list(islice(A121741_gen(),20)) # Chai Wah Wu, Jul 03 2023

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = ab(c+d) = (a+b)cd.

Original entry on oeis.org

72, 144, 168, 360, 450, 480, 576, 864, 990, 1152, 1200, 1344, 1404, 1568, 1600, 1800, 1944, 2040, 2160, 2520, 2646, 2880, 3150, 3360, 3600, 3780, 3840, 3888, 4050, 4536, 4608, 4800, 5184, 5400, 5520, 5880, 5940, 6720, 6912, 7056, 7200, 7350, 7800, 7920, 7938, 8550, 8640, 8694, 8712, 8976, 9000, 9216, 9408, 9450, 9504, 9600, 9720, 10416, 10752, 11232, 11550, 11760

Offset: 1

Max Alekseyev, Jan 20 2024

nonn

Such numbers m correspond to pairs of equal Egyptian fractions of length 2, since a*b*(c+d) = (a+b)*c*d is equivalent to 1/a + 1/b = 1/c + 1/d.
Numbers of the form t * k^3, where t is a term of A369334 and k is a positive integer.
If m belongs to this sequence, then so does m*k^3 for any positive integer k.

			72 is a term since 72 = 3*3*(2+6) = (3+3)*2*6.

Cf. A088915, A369334, A371721.

{ find_abcd(m) = my(r); fordiv(m,a, if(2*a*a>m, break); fordiv(m\a,b, if(4*a^2*b^2 > m*(a+b), break); if(bdenominator(x)==1, nfroots(,x^2 - m\a\b*x + m\(a+b))); if(#r==1, r=[r[1],r[1]]); if(#r==2, return([a,b,r[1],r[2]])); )); []; }

A371721 Numbers representable in the form uv(u+v) for u >= v >= 0 in at least two ways.

Original entry on oeis.org

0, 30, 210, 240, 390, 420, 462, 810, 880, 1008, 1020, 1056, 1122, 1190, 1482, 1680, 1920, 1980, 2070, 2100, 2310, 2970, 3120, 3360, 3696, 3750, 4160, 4290, 4320, 4830, 4914, 5460, 5670, 6006, 6090, 6270, 6480, 6630, 7040, 7380, 7440, 7770, 8064, 8160, 8190, 8448, 8580, 8976, 9120, 9240, 9520, 9900, 10290, 10530, 10640, 11340, 11856, 12210, 12432, 12474, 13110

Offset: 1

Max Alekseyev, Apr 04 2024

nonn

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

Subsequence of A088915.

Cf. A369333, A369334.

N:= 10^5: # for terms <= N
V:= Vector(N):
for v from 1 while 2*v^3 <= N do
  for u from v do
    x:= u*v*(u+v);
    if x > N then break fi;
    V[x]:= V[x]+1
od od:
[0,op(select(t -> V[t] >= 2, [$1..N]))]; # Robert Israel, Dec 15 2024

A363052 Integers m for which there exist positive integers j, k such that jk(j+k) = m^2.

Original entry on oeis.org

4, 18, 24, 32, 36, 50, 60, 108, 140, 144, 150, 192, 252, 256, 288, 300, 360, 392, 400, 480, 486, 500, 540, 588, 648, 780, 816, 864, 882, 900, 972, 1008, 1014, 1050, 1120, 1152, 1156, 1176, 1200, 1350, 1372, 1452, 1536, 1620, 1764, 1800, 1848, 2016, 2040, 2048, 2178

Offset: 1

Zhining Yang, May 15 2023

All terms are even.

			24 is a term: j*k*(j+k) = 24^2 for j=2, k=16.

Cf. A088915.

Select[2*Range@500,
 Length@Select[Table[(Sqrt[b^2 + 4 #^2/b] - b)/2, {b, #}], IntegerQ] >
    0 &]
Select[Union@
  Flatten@Table[Sqrt[a*b (a + b)], {a, 1, 80}, {b, a, 500}],
 IntegerQ[#] && # < 1000 &]

from itertools import count, islice
from sympy import integer_nthroot, divisors
def A363052_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        for k in divisors(m**2,generator=True):
            p, q = integer_nthroot(k**4+(k*m**2<<2),2)
            if q:
                a, b = divmod(p-k**2,k<<1)
                if a > 0 and not b:
                    yield m
                    break
A363052_list = list(islice(A363052_gen(),20)) # Chai Wah Wu, Jul 03 2023

cp's OEIS Frontend

A121741 Dimensions of the irreducible representations of the simple Lie algebra of type A2 (equivalently, the group SL3) over the complex numbers, listed in increasing order.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Python

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = ab(c+d) = (a+b)cd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A371721 Numbers representable in the form uv(u+v) for u >= v >= 0 in at least two ways.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A363052 Integers m for which there exist positive integers j, k such that jk(j+k) = m^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

cp's OEIS Frontend

A121741 Dimensions of the irreducible representations of the simple Lie algebra of type A2 (equivalently, the group SL3) over the complex numbers, listed in increasing order.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Python

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = a*b*(c+d) = (a+b)*c*d.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A371721 Numbers representable in the form u*v*(u+v) for u >= v >= 0 in at least two ways.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A363052 Integers m for which there exist positive integers j, k such that j*k*(j+k) = m^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = ab(c+d) = (a+b)cd.

A371721 Numbers representable in the form uv(u+v) for u >= v >= 0 in at least two ways.

A363052 Integers m for which there exist positive integers j, k such that jk(j+k) = m^2.