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A057896 Nonnegative numbers that can be written as m^k - m (with m and k nonnegative) in more than one way.

%I A057896 #49 May 24 2019 02:04:22
%S A057896 0,6,30,210,240,2184,8190,78120,24299970
%N A057896 Nonnegative numbers that can be written as m^k - m (with m and k nonnegative) in more than one way.
%C A057896 The next term, if it exists, is at least 2*10^17. - _David Wasserman_, May 01 2002
%C A057896 a(10) > 10^24, if it exists. The only numbers below 10^24 that can be written as m^k+m in more than one way are 30 = 5^2+5 = 3^3+3 and 130 = 5^3+5 = 2^7+2. - _Giovanni Resta_, Jun 21 2018
%C A057896 Conjectured to be finite and complete by Bennett (2001).
%H A057896 Michael Bennett, <a href="https://doi.org/10.4153/CJM-2001-036-6">On some exponential equations of S. S. Pillai</a>, Canad. J. Math. 53 (2001), 897-922.
%H A057896 Brady Haran and Matt Parker, <a href="https://youtu.be/99stb2mzspI">Why 1980 was a great year to be born... but 2184 will be better</a>, Numberphile video (2015).
%H A057896 Dana Mackenzie, <a href="http://math.colgate.edu/~integers/s33/s33.Abstract.html">2184: An Absurd (and Adsurd) Tale</a>, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33. See Conjecture 3.
%e A057896 30 is in the sequence since 30 = 2^5 - 2 = 6^2 - 6;
%e A057896 2184 is in the sequence since 2184 = 3^7 - 3 = 13^3 - 13.
%p A057896 res:= {0}:
%p A057896 for k from 3 to 60 do
%p A057896   for m from 2 while m^k-m < 2^60 do
%p A057896      x:= m^k-m;
%p A057896      if assigned(R[x]) or issqr(4*x+1) then res:= res union {x}
%p A057896      else R[x]:= [m,k]
%p A057896      fi
%p A057896   od
%p A057896 od:
%p A057896 res; # _Robert Israel_, Oct 07 2015
%Y A057896 Cf. A000918, A057895, A308324.
%K A057896 more,nonn
%O A057896 1,2
%A A057896 _Henry Bottomley_, Sep 26 2000
%E A057896 More terms from _Jud McCranie_, Oct 01 2000
%E A057896 Offset corrected by _Joerg Arndt_, Oct 07 2015