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

14, 88, 65548, 33554459, 387420510, 1099511627800, 35184372088855, 3656158440063002, 459986536544739960976836

Offset

1,1

Comments

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)

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)

Links

Table of n, a(n) for n=1..9.

Example

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.

Prog

(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])

Crossrefs

Cf. A254034.

Sequence in context: A116343 A259473 A202785 * A034544 A248060 A186257

Adjacent sequences: A255532 A255533 A255534 * A255536 A255537 A255538

Keyword

nonn,more

Author

Alex Ratushnyak, Feb 24 2015

Extensions

a(5)-a(8) from Lars Blomberg, May 19 2015

a(9) from Chai Wah Wu, May 18 2021

Status

approved