A242100 Numbers of the form m = b^i + b^j, where b > 1 and i > j > 0.

6, 10, 12, 18, 20, 24, 30, 34, 36, 40, 42, 48, 56, 66, 68, 72, 80, 84, 90, 96, 108, 110, 130, 132, 136, 144, 150, 156, 160, 182, 192, 210, 222, 240, 246, 252, 258, 260, 264, 270, 272, 288, 306, 320, 324, 342, 350, 380, 384, 392, 420, 462, 506, 514, 516, 520

Offset: 1

Author

Hieronymus Fischer, May 04 2014

Keywords

• nonn

Comments

If m is a term, then there is a base b > 1 such that the base-b representation of m has digital sum = 2.

The base b for which m = b^i + b^j is not uniquely determined. Example: 12 = 2^3+2^2 = 3^2 +3^1.

Examples

			a(1)    = 6, since 2 = 2^2 + 2^1.
a(7)    = 30, since 30 = 3^3 + 3^1.
a(10)   = 40.
a(10^2) = 1722.
a(10^3) = 377610.
a(10^4) = 70635620.
a(10^5) = 8830078992.
a(10^6) = 951958292172.
a(10^7) = 97932587392010.
a(10^8) = 9908034917287656.
a(10^9) = 995834160614903742.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597, A018900, A239709, A239710.

Programs

Smalltalk

distinctPowersWithOffset: d
  "Answers an array which holds the first n numbers of the form b^i + b^j + d, i>j>0, where b is any natural number > 1, d is any integer number, and n is the receiver (d=0 for this sequence).
  Usage: n distinctPowersWithOffset: 0
  Answer: #(6 10 12 ...) [first n terms]"
  | n terms m |
  terms := SortedCollection new.
  n := self.
  m := n squared max: 20.
  terms := m floorDistinctPowersWithOffset: d.
  ^terms copyFrom: 1 to: n
----------
floorDistinctPowersWithOffset: d
  "Answers an array which holds the numbers < n of the form b^i + b^j + d, i>j>0, where b is any natural number > 1, d is any integer number, and n is the receiver (d=0 for this sequence).
  Usage: n floorDistinctPowersWithOffset: 0
  Answer: #(6 10 12 18 ...) [all terms < n]"
  | bmax p q n m terms a |
  terms := OrderedCollection new.
  n := self.
  bmax := ((4 * (n - d) + 1) sqrtTruncated - 1) // 2.
  2 to: bmax
    do:
         [:b |
         p := b * b.
         q := b.
         a := p + q + d.
         [a < n] whileTrue:
                   [[q < p and: [a < n]] whileTrue:
                            [terms add: a.
                            q := b * q.
                            a := p + q + d].
                   p := b * p.
                   q := b.
                   a := p + q + d]].
  ^terms asSet asOrderedCollection sorted

Formula

a(n) < n^2 for n > 4.

lim a(n)/n^2 = 1, for n --> infinity.