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.

Showing 1-2 of 2 results.

A255535 Numbers representable as both b^c + b + c and x^y + x - y, where b, c, x, y are integers greater than 1.

Original entry on oeis.org

14, 88, 65548, 33554459, 387420510, 1099511627800, 35184372088855, 3656158440063002, 459986536544739960976836, 1237940039285380274899124273, 6362685441135942358474828762538534230890216378
Offset: 1

Views

Author

Alex Ratushnyak, Feb 24 2015

Keywords

Comments

From Chai Wah Wu, May 17 2021: (Start)
Sequence is infinite.
If a, b > 1 and b^a-b == 0 mod a+1 then b^c+b+c is a term for c = ab(b^(a-1)-1)/(a+1), y = c/a, x = b^a.
If b > 1 and b <> 2 mod 3, then b^(2b(b-1)/3)+b(2b+1)/3 is a term.
If b > 2, then b^((b-1)(b^(b-2)-1)) + b + (b-1)(b^(b-2)-1) is a term. (End)
From Chai Wah Wu, May 18 2021: (Start)
Either c>=3 or y>=3. If c=y=2, we get b^2+b+2=x^2+x-2, i.e. (x-b)(x+b+1) = 4. Since x>1 and b>1, x+b+1>4, a contradiction.
This allows for a faster search algorithm by assuming c>=3 and y>=3. The cases c=2 and y>=3 can be dealt with by picking y>=3 and solving for b in the quadratic equation b^2+b+2=x^y+x-y. Similarly for c>=3 and y=2. This approach was used to confirm a(9). (End)
For n >= 3, we have max(c,y) >= 5. First note that c == y (mod 2). Case (c,y) = (3,3) implies (x-b)|6 and leads to quadratic equations with no integer roots. Case (c,y) = (4,2) corresponds to a quartic curve and has the only solution (b,x) = (3,9) giving a(2)=88, while case (c,y) = (2,4) has the only solution (b,x) = (3,2) giving a(1)=14. Finally, case (c,y) = (4,4) implies (x-b)|8 and leads to cubic equations with no integer roots. - Max Alekseyev, Feb 10 2025

Examples

			a(1) = 14 = 3^2 + 3 + 2 = 2^4 + 2 - 4.
a(2) = 88 = 3^4 + 3 + 4 = 9^2 + 9 - 2.
a(3) = 65548 = 4^8 + 4 + 8 = 16^4 + 16 - 4.
a(4) = 33554459 = 2^25 + 2 + 25 = 32^5 + 32 - 5.
a(5) = 387420510 = 3^18 + 3 + 18 = 27^6 + 27 - 6.
a(6) = 1099511627800 = 4^20 + 4 + 20 = 32^8 + 32 - 8.
a(7) = 35184372088855 = 8^15 + 8 + 15 = 32^9 + 32 - 9.
a(8) = 3656158440063002 = 6^20 + 6 + 20 = 36^10 + 36 - 10.
From _Michael S. Branicky_, May 15 2021: (Start)
The following are terms:
459986536544739960976836      =  7^28 +  7 + 28 =  49^14 +  49 - 14,
1237940039285380274899124273  =  4^45 +  4 + 45 =  64^15 +  64 - 15,
6362685...0216378 (46 digits) =  9^48 +  9 + 48 =  81^24 +  81 - 24,  and
1000000...0000070 (61 digits) = 10^60 + 10 + 60 = 100^30 + 100 - 30. (End)

Crossrefs

Cf. A235368, A254034.

Programs

  • Python
    TOP = 100000000
a = [0]*TOP
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
    a[k]=1
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
    if k>=0: a[k]|=2
print([n for n in range(TOP) if a[n]==3])

Extensions

a(5)-a(8) from Lars Blomberg, May 19 2015
a(9) from Michael S. Branicky confirmed by Chai Wah Wu, May 18 2021
a(10)-a(11) from Michael S. Branicky confirmed by Max Alekseyev, Mar 02 2025

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.

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

Views

Author

Alex Ratushnyak, Mar 07 2015

Keywords

Comments

Intersection of A253775, A254671, A255265.

Examples

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

Links

Crossrefs

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

Programs

  • PARI
    \\ See Corneth link
  • Python
    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)])

Extensions

More terms from David A. Corneth, Oct 18 2024
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