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%I #32 Jun 19 2023 07:09:20
%S 0,1,7,17,28,79,118,192,399,431,513,924,1844,1927,2800,3952,6049,7849,
%T 8023,13983,16188,18954,32543,58049,61318,61440,65280,130783,162287,
%U 175816,255583,261820,357857,523927,529713,1038576,1048176
%N Nonnegative numbers of the form x^y - y^x, for x,y > 1.
%C Pillai proved that there are ~ 0.5 * (log x)^2/(log log x)^2 terms of this sequence up to x. - _Charles R Greathouse IV_, Jul 20 2017
%C Conjecture: For d > 11, 10^d - d^10 is the largest (base-ten) d-digit term. - _Hans Havermann_, Jun 12 2023
%D S. S. Pillai, On the indeterminate equation x^y - y^x = a, Journal Annamalai University 1, Nr. 1, (1932), pp. 59-61. Cited in Waldschmidt 2009.
%H Charles R Greathouse IV, <a href="/A045575/b045575.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H Michel Waldschmidt, <a href="https://arxiv.org/abs/0908.4031">Perfect Powers: Pillai's works and their developments</a>, arXiv:0908.4031 [math.NT], 2009.
%p N:= 10^8: # to get all terms <= N
%p A:= (0,1):
%p for x from 2 while x^(x+1) - (x+1)^x <= N do
%p for y from x+1 do
%p z:= x^y - y^x;
%p if z > N then break
%p elif z > 0 then A:=A,z;
%p fi
%p od od:
%p {A}; # _Robert Israel_, Aug 20 2014
%t Union[Flatten[Table[If[a^b-b^a>-1&&a^b-b^a<10^6*2,a^b-b^a],{a,1,123},{b,a,144}]]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 26 2008 *)
%t nn=10^50; n=1; Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num<nn, Sow[num]; While[k++; num=n^k-k^n; num<nn, Sow[num]]]][[2,1]]]
%o (PARI) list(lim)=my(v=List([0]),t); for(x=2,max(logint(lim\=1,2)+1,6), for(y=2,x-1, t=abs(x^y-y^x); if(t<=lim&&t, listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, Jul 20 2017
%Y Cf. A076980.
%K easy,nonn
%O 1,3
%A _Erich Friedman_
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