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%I #14 Oct 21 2020 02:50:45
%S 2,5,7,12,14,23,24,27,34,47,58,61,62,77,79,98,119,121,122,142,167,194,
%T 213,223,238,248,252,254,287,322,340,359,398,439,482,503,509,527,574,
%U 621,623,674,723,726,727,782,839,898,959,997,1014,1019,1022,1087,1154
%N Numbers which can be written as m^k-k, with m, k > 1.
%C It may be that positive integers can be written as m^k-k (with m and k > 1) in at most one way [checked up to 10000].
%C All numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x-y for x > y > 1 and b > a > 1, which appears to have no solutions. - _T. D. Noe_, Oct 06 2004
%H Andrew Howroyd, <a href="/A057897/b057897.txt">Table of n, a(n) for n = 1..10000</a>
%t nLim=1000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst] (* _T. D. Noe_, Oct 06 2004 *)
%o (PARI) ok(n)={my(e=2); while(2^e <= n+e, if(ispower(n+e, e), return(1)); e++); 0} \\ _Andrew Howroyd_, Oct 20 2020
%o (PARI) upto(lim)={my(p=logint(lim,2)); while(logint(lim+p+1,2)>p, p++); Vec(Set(concat(vector(p-1, e, e++; vector(sqrtnint(lim+e,e)-1, m, (m+1)^e-e)))))} \\ _Andrew Howroyd_, Oct 20 2020
%Y Cf. A000325, A008865, A024024, A024037, A024050, A057895, A057898, A057899.
%Y Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).
%K nonn
%O 1,1
%A _Henry Bottomley_, Sep 26 2000
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