A255265 -id:A255265 - cp's OEIS frontend

A255267 Numbers representable as both xy(x+y) and b*c+b+c, where b>=c>1 and x>=y>1.

48, 54, 84, 120, 128, 160, 264, 286, 308, 324, 384, 390, 468, 510, 560, 624, 686, 714, 720, 798, 840, 884, 912, 960, 1024, 1056, 1134, 1140, 1190, 1224, 1254, 1280, 1330, 1350, 1386, 1440, 1456, 1500, 1512, 1584, 1650, 1672, 1680, 1710, 1748, 1794, 1798, 1820, 1890

Offset: 1

Alex Ratushnyak, Feb 19 2015

nonn

Intersection of A254671 and A255265.

The subsequence of squares begins: 324, 1024, 2500, 3600, 11664, 19600, 20736, 36864, 63504, 82944, 129600, 153664, 230400, 236196, 250000, 291600, 345744, 419904, 777924, 810000, 944784.

			a(3) = 84 = 4*3*(4+3) = 16*4 + 16 + 4.

David A. Corneth, Table of n, a(n) for n = 1..12033 (terms <= 2*10^6)
David A. Corneth, PARI program

Cf. A254671, A255265.

PARI
```
\\ See Corneth link
```

TOP = 100000
a = [0]*TOP
b = [0]*TOP
for y in range(2,TOP//2):
  for x in range(y,TOP//2):
    k = x*y*(x+y)
    if k>=TOP: break
    a[k]+=1
for y in range(2,TOP//2):
  for x in range(y,TOP//2):
    k = x*y+(x+y)
    if k>=TOP: break
    b[k]+=1
print([n for n in range(TOP) if a[n]>0 and b[n]>0])

A255804 Numbers representable as xy(x+y), b*c+b+c, and d^e+d+e, where d>1, e>1, b>=c>1 and x>=y>1.

Original entry on oeis.org

264, 308, 8192, 16400, 88508, 236684, 504812, 12127808, 22491308, 82310258, 227240552, 385278014, 1069061114, 2363758544, 2591166314, 2985365684, 3310448834, 4042988642, 4791339182, 5712714308, 7553782658, 8626601522, 12494656622, 14498688512, 15165306758, 15445891244

Offset: 1

Alex Ratushnyak, Mar 07 2015

nonn

Intersection of A253775, A254671, A255265.

			a(2) = 308 = 17^2 + 17 + 2 = 7 * 4 * (7 + 4) = 102 * 2 + 102 + 2.

David A. Corneth, Table of n, a(n) for n = 1..160
David A. Corneth, PARI program

Cf. A255267, A254034, A255265, A254671, A253775.

PARI
```
\\ See Corneth link
```

TOP = 100000000
a = [0]*TOP
c = []
for y in range(2, TOP//2):
  if 2**y + 2 + y >= TOP: break
  for x in range(2, TOP//2):
    k = x**y+(x+y)
    if k>=TOP: break
    c.append(k)
for y in range(2, TOP//2):
  if 2*y*y*y >= TOP: break
  for x in range(y, TOP//2):
    k = x*y*(x+y)
    if k>=TOP: break
    a[k]=1
for y in range(2, TOP//2):
  if y*(y+2) >= TOP: break
  for x in range(y, TOP//2):
    k = x*y+(x+y)
    if k>=TOP: break
    a[k]|=2
    # if a[k]==3 and (k in c): print(k, end=', ')
print([n for n in range(TOP) if a[n]==3 and (n in c)])

More terms from David A. Corneth, Oct 18 2024

A277980 a(n) = 12n^2 + 18n.

Original entry on oeis.org

0, 30, 84, 162, 264, 390, 540, 714, 912, 1134, 1380, 1650, 1944, 2262, 2604, 2970, 3360, 3774, 4212, 4674, 5160, 5670, 6204, 6762, 7344, 7950, 8580, 9234, 9912, 10614, 11340, 12090, 12864, 13662, 14484, 15330, 16200, 17094, 18012, 18954, 19920

Offset: 0

Emeric Deutsch, Nov 08 2016

For n>=3, a(n) is the second Zagreb index of the double-wheel graph DW[n]. The second Zagreb index of a simple connected graph g is the sum of the degree products d(i) d(j) over all edges ij of g.
The double-wheel graph DW[n] consists of two cycles C[n], whose vertices are connected to an additional vertex.
The M-polynomial of the double-wheel graph DW[n] is M(DW[n],x,y) = 2*n*x^3*y^3 + 2*n*x^3*y^{2*n}.

			a(3) = 162. Indeed, the double-wheel graph DW[3] has 6 edges with end-point degrees 3,3 and 6 edges with end-point degrees 3,6. Then the second Zagreb index is 6*9 + 6*18 = 162.

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014106, A152746, A154105, A277979.
First bisection of A277978.
After 0, subsequence of A255265.

[12*n^2+18*n: n in [0..40]]; // Vincenzo Librandi, Nov 09 2016

Maple
```
seq(12*n^2+18*n, n = 0 .. 50);
```

Table[12 n^2 + 18 n, {n, 0, 45}] (* Vincenzo Librandi, Nov 09 2016 *)

a(n)=12*n^2+18*n \\ Charles R Greathouse IV, Nov 09 2016

G.f.: 6*x*(5-x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 6*A014106(n).
a(n) = A152746(n+1) - 6 = A154105(n) - 7. - Omar E. Pol, May 08 2018

A255266 Numbers representable in more than one way as xy(x+y), where x>=y>1.

Original entry on oeis.org

240, 390, 810, 880, 1008, 1020, 1680, 1920, 2100, 2310, 2970, 3120, 3360, 3696, 3750, 4320, 4914, 5460, 5670, 6090, 6270, 6480, 6630, 7040, 7380, 7440, 7770, 8064, 8160, 8190, 8448, 8580, 8976, 9240, 9520, 10290, 10530, 10640, 11340, 11856, 12474, 13440, 13776, 14040

Offset: 1

Alex Ratushnyak, Feb 19 2015

nonn

			a(3) = 810 = 15*3*(15+3) = 9*6*(9+6).

Cf. A255265, A081779.

TOP = 100000
a = [0]*TOP
for y in range(2, TOP//2):
  for x in range(y, TOP//2):
    k = x*y*(x+y)
    if k>=TOP: break
    a[k]+=1
print([n for n in range(TOP) if a[n]>1])

A376641 a(n) is the first positive integer that has exactly n representations as x * y * (x + y) where x >= y > 1.

Original entry on oeis.org

1, 16, 240, 1680, 23760, 18480, 498960, 24386670, 3991680, 57805440, 618068880, 195093360, 4944551040, 1560746880, 12485975040, 99887800320, 2696970608640, 195093360000, 1560746880000, 5267520720000

Offset: 0

Robert Israel, Sep 30 2024

			a(1) = 16 =  2 * 2 * (2 + 2).
a(2) = 240 = 10 * 2 * (10 + 2)
           = 6 * 4 * (6 + 4).
a(3) = 1680 = 28 * 2 * (28 + 2)
            = 16 * 5 * (16 + 5)
            = 14 * 6 * (14 + 6).
a(4) = 23760 = 108 * 2 * (108 + 2)
           =  60 * 6 * (60 + 6)
           =  44 * 10 * (44 + 10)
           =  33 * 15 * (33 + 15).

Cf. A255265.

N:= 10^8: # for terms <= N
V:= Vector(N,datatype=integer[1]):
for y from 2 to floor((N/2)^(1/3)) do
  for x from y do
    v:= x*y *(x+y);
    if v > N then break fi;
    V[v]:= V[v]+1
od od:
m:= max(V): R:= Array(0..m): count:= 0:
for i from 1 to N while count < m+1 do
  v:= V[i];
  if R[v] = 0 then R[v]:= i; count:= count+1 fi
od:
convert(R,list);

a(12)-a(19) from Chai Wah Wu, Oct 01 2024

A382953 Numbers with at least one factorization for which the factors can be partitioned into 2 or more distinct subsets with equal sums.

Original entry on oeis.org

16, 30, 48, 54, 64, 70, 72, 84, 96, 120, 126, 128, 144, 160, 162, 180, 192, 198, 210, 216, 240, 243, 250, 252, 256, 264, 270, 280, 286, 288, 300, 308, 320, 324, 330, 336, 360, 378, 384, 390, 396, 400, 420, 432, 440, 448, 462, 468, 480, 486, 495, 504, 510, 512

Offset: 1

Charles L. Hohn, Apr 09 2025

nonn

Here "distinct" means that no partition contains the same subset of factors, e.g. 4 is not a term because {2} == {2}.
Because 2 + 2 = 2 * 2 = 4, many terms have multiple instances that differ only by factors {2, 2} vs. {4}, except in some cases where such substitutions would create indistinct subsets, e.g. while 16 is a term for partition set {{2, 2}, {4}}, {{2, 2}, {2, 2}} and {{4}, {4}} do not count as additional instances.
For primes p and integers x >= 0, p^(p+2+2x) and p^(2p+3+x) are terms.
For integers x and y >= 0, (4x+4)^(y+2) and (4x+6)^(y+3) are terms.
First few terms with record counts of instances: 16 (1 instance), 48 (2), 120 (3), 240 (6), 576 (8), 720 (9), 768 (12).
If k is a term, then 4k is also a term. - Ivan N. Ianakiev, Apr 10 2025

			a(1) = 16: 2 * 2 * 4 = 16 and 2 + 2 = 4.
a(2) = 30: 2 * 3 * 5 = 30 and 2 + 3 = 5.
a(3) = 48: 2 * 2 * 2 * 6 = 48 and 2 + 2 + 2 = 6, and also 2 * 4 * 6 = 48 and 2 + 4 = 6.
a(5) = 64: 2 * 2 * 2 * 2 * 4 = 64 and 2 + 2 + 2 = 2 + 4.
a(39) = 384: 2 * 2 * 2 * 2 * 4 * 6 = 384 and 2 + 2 + 2 = 2 + 4 = 6 (plus 4 other instances).

Cf. A083207, A322657, A255265 (subsequence).

ok[n_]:=Catch@ Block[{t, d=Divisors@n,f}, f[y_]:=Block[{L={}, r}, r[x_,m_,c_]:= If[x==1, AppendTo[L,c], r[x/#, #, Append[c,#]]& /@ Select[ Divisors@x, #>=m&];]; f[y,2,{}]; L]; Do[t=Plus@@@ s[d[[i]]]; If[d[[i]]^2!=n, Intersection[t, Plus@@@ s[n/d[[i]]]] != {} && Throw@True, Sort@t != Union@t && Throw@True],{i, 2, Ceiling[ Length@d/2]}]; False]; Select[Range@ 512,ok] (* Giovanni Resta, Apr 10 2025 *)

a382953_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(#f && r==1, my(c=0, s=vecsum(Vec(f)), d=divisors(s)); for(i=2, #d, my(z=s/d[i]); if(z1, next); listput(f, d); c+=a382953_count(x, f); listpop(f)); return(c)}
a382953_part(f, z, rvs=0, v=List())={my(c=0); if(#v==#f[2], if(sum(i=1, #v, f[1][i]*v[i])

cp's OEIS Frontend

A255267 Numbers representable as both x*y*(x+y) and b*c+b+c, where b>=c>1 and x>=y>1.

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A255804 Numbers representable as x*y*(x+y), b*c+b+c, and d^e+d+e, where d>1, e>1, b>=c>1 and x>=y>1.

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A277980 a(n) = 12*n^2 + 18*n.

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Magma

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Formula

A255266 Numbers representable in more than one way as x*y*(x+y), where x>=y>1.

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Author

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Examples

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Programs

Python

A376641 a(n) is the first positive integer that has exactly n representations as x * y * (x + y) where x >= y > 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

A382953 Numbers with at least one factorization for which the factors can be partitioned into 2 or more distinct subsets with equal sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A255267 Numbers representable as both xy(x+y) and b*c+b+c, where b>=c>1 and x>=y>1.

A255804 Numbers representable as xy(x+y), b*c+b+c, and d^e+d+e, where d>1, e>1, b>=c>1 and x>=y>1.

A277980 a(n) = 12n^2 + 18n.

A255266 Numbers representable in more than one way as xy(x+y), where x>=y>1.