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 A023056 a(n) is least k such that k and k+n are adjacent nontrivial powers of positive integers, or 0 if no such k apparently exists. 5

%I

%S 8,25,1,4,27,0,9,97336,16,2187,3125,2197,36,0,49,128,64,225,81,196,

%T 100,0,2025,1000,144,42849,169,484,0,6859,0,7744,256,0,289,1728,

%U 14348907,1331,361,2704,400,0,441,0,9216,0,529,21904,576,0,625,0,676,0,729,5776,784,0

%N a(n) is least k such that k and k+n are adjacent nontrivial powers of positive integers, or 0 if no such k apparently exists.

%C Searching up to 10^22, the largest term for n <= 1000 is a(618) = 421351^3 = 74805251419106551. - T. D. Noe, Apr 21 2011

%H Robert G. Wilson v, <a href="/A023056/b023056.txt">Table of n, a(n) for n = 1..1000 </a>. [From _Robert G. Wilson v_, May 29 2009]

%t nextPerfectPowers[n_] := Block[{k = n + 1}, While[GCD @@ Last /@ FactorInteger@ k == 1, k++ ]; k]; t = Table[0, {100}]; t[[3]] = 1; m = 0; While[m < 14400000, n = nextPerfectPowers@ m; d = n - m; If[d < 100 && t[[d]] == 0, t[[d]] = m; Print[{d, m}]]; m = n]; t [From _Robert G. Wilson v_, May 29 2009]

%t (* checked against *) mx = 14400000; pp = Union[ Join[{1}, Flatten[ Table[n^i, {n, 2, Sqrt@mx}, {i, 2, Log[n, mx]}]]]]; d = Rest@ pp - Most@ pp; pp[[ # ]] & /@ Flatten[ Table[ Position[d, n, 1, 1], {n, 56}] /. {{} -> {0}}] /. {List -> 0} [From _Robert G. Wilson v_, May 29 2009]

%Y Cf. A189117 (conjectured number of pairs of consecutive perfect powers differing by n).

%K nonn

%O 1,1

%A _David W. Wilson_

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