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

%I A242100 #13 May 14 2014 19:43:27
%S A242100 6,10,12,18,20,24,30,34,36,40,42,48,56,66,68,72,80,84,90,96,108,110,
%T A242100 130,132,136,144,150,156,160,182,192,210,222,240,246,252,258,260,264,
%U A242100 270,272,288,306,320,324,342,350,380,384,392,420,462,506,514,516,520
%N A242100 Numbers of the form m = b^i + b^j, where b > 1 and i > j > 0.
%C A242100 If m is a term, then there is a base b > 1 such that the base-b representation of m has digital sum = 2.
%C A242100 The base b for which m = b^i + b^j is not uniquely determined. Example: 12 = 2^3+2^2 = 3^2 +3^1.
%H A242100 Hieronymus Fischer, <a href="/A242100/b242100.txt">Table of n, a(n) for n = 1..10000</a>
%F A242100 a(n) < n^2 for n > 4.
%F A242100 lim a(n)/n^2 = 1, for n --> infinity.
%e A242100 a(1)    = 6, since 2 = 2^2 + 2^1.
%e A242100 a(7)    = 30, since 30 = 3^3 + 3^1.
%e A242100 a(10)   = 40.
%e A242100 a(10^2) = 1722.
%e A242100 a(10^3) = 377610.
%e A242100 a(10^4) = 70635620.
%e A242100 a(10^5) = 8830078992.
%e A242100 a(10^6) = 951958292172.
%e A242100 a(10^7) = 97932587392010.
%e A242100 a(10^8) = 9908034917287656.
%e A242100 a(10^9) = 995834160614903742.
%o A242100 (Smalltalk)
%o A242100 distinctPowersWithOffset: d
%o A242100   "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).
%o A242100   Usage: n distinctPowersWithOffset: 0
%o A242100   Answer: #(6 10 12 ...) [first n terms]"
%o A242100   | n terms m |
%o A242100   terms := SortedCollection new.
%o A242100   n := self.
%o A242100   m := n squared max: 20.
%o A242100   terms := m floorDistinctPowersWithOffset: d.
%o A242100   ^terms copyFrom: 1 to: n
%o A242100 ----------
%o A242100 floorDistinctPowersWithOffset: d
%o A242100   "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).
%o A242100   Usage: n floorDistinctPowersWithOffset: 0
%o A242100   Answer: #(6 10 12 18 ...) [all terms < n]"
%o A242100   | bmax p q n m terms a |
%o A242100   terms := OrderedCollection new.
%o A242100   n := self.
%o A242100   bmax := ((4 * (n - d) + 1) sqrtTruncated - 1) // 2.
%o A242100   2 to: bmax
%o A242100     do:
%o A242100          [:b |
%o A242100          p := b * b.
%o A242100          q := b.
%o A242100          a := p + q + d.
%o A242100          [a < n] whileTrue:
%o A242100                    [[q < p and: [a < n]] whileTrue:
%o A242100                             [terms add: a.
%o A242100                             q := b * q.
%o A242100                             a := p + q + d].
%o A242100                    p := b * p.
%o A242100                    q := b.
%o A242100                    a := p + q + d]].
%o A242100   ^terms asSet asOrderedCollection sorted
%Y A242100 Cf. A001597, A018900, A239709, A239710.
%K A242100 nonn
%O A242100 1,1
%A A242100 _Hieronymus Fischer_, May 04 2014