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.

8, 25, 1, 4, 27, 0, 9, 97336, 16, 2187, 3125, 2197, 36, 0, 49, 128, 64, 225, 81, 196, 100, 0, 2025, 1000, 144, 42849, 169, 484, 0, 6859, 0, 7744, 256, 0, 289, 1728, 14348907, 1331, 361, 2704, 400, 0, 441, 0, 9216, 0, 529, 21904, 576, 0, 625, 0, 676, 0, 729, 5776, 784, 0

Offset

1,1

Comments

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

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Mathematica

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 (* Robert G. Wilson v, May 29 2009 *)
(* 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} (* Robert G. Wilson v, May 29 2009 *)

Crossrefs

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

Cf. A103954. (the powerful (A001694) analogous sequence).

Sequence in context: A208400 A103953 A076444 * A103954 A217012 A200838

Adjacent sequences: A023053 A023054 A023055 * A023057 A023058 A023059

Keyword

nonn

Author

David W. Wilson

Status

approved