A255535 - OEIS

%I #50 Mar 03 2025 13:24:40

%S 14,88,65548,33554459,387420510,1099511627800,35184372088855,

%T 3656158440063002,459986536544739960976836,

%U 1237940039285380274899124273,6362685441135942358474828762538534230890216378

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

%C From _Chai Wah Wu_, May 17 2021: (Start)

%C Sequence is infinite.

%C 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.

%C If b > 1 and b <> 2 mod 3, then b^(2b(b-1)/3)+b(2b+1)/3 is a term.

%C If b > 2, then b^((b-1)(b^(b-2)-1)) + b + (b-1)(b^(b-2)-1) is a term. (End)

%C From _Chai Wah Wu_, May 18 2021: (Start)

%C 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.

%C 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)

%C 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

%e a(1) = 14 = 3^2 + 3 + 2 = 2^4 + 2 - 4.

%e a(2) = 88 = 3^4 + 3 + 4 = 9^2 + 9 - 2.

%e a(3) = 65548 = 4^8 + 4 + 8 = 16^4 + 16 - 4.

%e a(4) = 33554459 = 2^25 + 2 + 25 = 32^5 + 32 - 5.

%e a(5) = 387420510 = 3^18 + 3 + 18 = 27^6 + 27 - 6.

%e a(6) = 1099511627800 = 4^20 + 4 + 20 = 32^8 + 32 - 8.

%e a(7) = 35184372088855 = 8^15 + 8 + 15 = 32^9 + 32 - 9.

%e a(8) = 3656158440063002 = 6^20 + 6 + 20 = 36^10 + 36 - 10.

%e From _Michael S. Branicky_, May 15 2021: (Start)

%e The following are terms:

%e 459986536544739960976836      =  7^28 +  7 + 28 =  49^14 +  49 - 14,

%e 1237940039285380274899124273  =  4^45 +  4 + 45 =  64^15 +  64 - 15,

%e 6362685...0216378 (46 digits) =  9^48 +  9 + 48 =  81^24 +  81 - 24,  and

%e 1000000...0000070 (61 digits) = 10^60 + 10 + 60 = 100^30 + 100 - 30. (End)

%o (Python)

%o TOP = 100000000

%o a = [0]*TOP

%o for y in range(2,TOP//2):

%o   if 2**y+2+y>=TOP: break

%o   for x in range(2,TOP//2):

%o     k = x**y+x+y

%o     if k>=TOP: break

%o     a[k]=1

%o for y in range(2,TOP//2):

%o   if 2**y+2-y>=TOP: break

%o   for x in range(2,TOP//2):

%o     k = x**y+x-y

%o     if k>=TOP: break

%o     if k>=0: a[k]|=2

%o print([n for n in range(TOP) if a[n]==3])

%Y Cf. A235368, A254034.

%K nonn,changed

%O 1,1

%A _Alex Ratushnyak_, Feb 24 2015

%E a(5)-a(8) from _Lars Blomberg_, May 19 2015

%E a(9) from _Michael S. Branicky_ confirmed by _Chai Wah Wu_, May 18 2021

%E a(10)-a(11) from _Michael S. Branicky_ confirmed by _Max Alekseyev_, Mar 2 2025