A251966 Numbers representable as both b^c - b + c and x^y + x - y, where b, c, x, y are integers greater than 1.

4, 14, 18, 27, 123, 256, 3125, 6556, 6566, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 95367431640610, 95367431640640, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177

Offset

1,1

Comments

A000312 except 1 is a subsequence.

Terms that are not in A000312: 14, 18, 123, 6556, 6566, 95367431640610, 95367431640640.

From Chai Wah Wu, May 18 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 <> 1 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.

For a(n) with n > 1, 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 the only solution is x=b=2 which corresponds to the term a(1) = 4.

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.

(End)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..30

Example

a(5) = 123 = 2^7 + 2 - 7 = 5^3 - 5 + 3.

Mathematica

Clear[b0, c0, x0, y0]; m = 100; max = 2^m; tb = Flatten[Table[b0[bc = b^c - b + c ] = b; c0[bc] = c; bc, {b, 2, m}, {c, 2, m}]]; tx = Flatten[Table[x0[xy = x^y + x - y] = x; y0[xy] = y; xy, {x, 2, m}, {y, 2, m}]]; inter = Intersection[Select[tb, # <= max &], Select[tx, # <= max &]]; Table[Print[n = inter[[k]], " b = ", b0[n], " c = ", c0[n], " x = ", x0[n], " y = ", y0[n]]; n, {k, Length[inter]}] (* Jean-François Alcover, Mar 23 2015 *)

Prog

(Python)
TOP = 10000000
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
    if k>=0: 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. A000312, A255535.

Sequence in context: A298126 A011859 A032825 * A022383 A336634 A045248

Adjacent sequences: A251963 A251964 A251965 * A251967 A251968 A251969

Keyword

nonn

Author

Alex Ratushnyak, Mar 21 2015

Status

approved