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 A340700 Lower of a pair of adjacent perfect powers, both with exponents > 2. 12
 27, 64, 125, 243, 1000, 1296, 2187, 50625, 59049, 194481, 279841, 456533, 614125, 3111696, 6434856, 22665187, 25411681, 38950081, 62742241, 96059601, 131079601, 418161601, 506250000, 741200625, 796594176, 1249198336, 2136719872, 2217342464, 5554571841, 5802782976 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It is conjectured that the intersection of A340700 and A340701 is empty, i.e., that no 3 immediately consecutive perfect powers with all exponents > 2 (A076467) exist. No counterexample < 3.4*10^30 was found. LINKS Hugo Pfoertner, Table of n, a(n) for n = 1..1670 StackExchange MathOverflow, Are there ever three perfect powers between consecutive squares? Answers by Gjergji Zaimi and Felipe Voloch (2011). Michel Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 27 Aug 2009. FORMULA a(n) = A340702(n)^A340704(n) = A340701(n) - A340706(n). EXAMPLE Initial terms of sequences A340700 .. A340706: a(n) = x^p, A340701(n) = A340703(n)^A340705(n) = y^q, A340706(n) = A340701(n) - a(n) = y^q - x^p. . n a(n) x ^ p A340701 y ^ q A340706 adjacent squares 1 27 = 3 ^ 3, 32 = 2 ^ 5, 5 5^2=25, 6^2=36 2 64 = 2 ^ 6, 81 = 3 ^ 4, 17 8^2=64, 9^2=81 3 125 = 5 ^ 3, 128 = 2 ^ 7, 3 11^2=121, 12^2=144 4 243 = 3 ^ 5, 256 = 2 ^ 8, 13 15^2=225, 16^2=256 5 1000 = 10 ^ 3, 1024 = 2 ^ 10, 24 31^2=961, 32^2=1024 6 1296 = 6 ^ 4, 1331 = 11 ^ 3, 35 36^2=1296, 37^2=1369 7 2187 = 3 ^ 7, 2197 = 13 ^ 3, 10 46^2=2116, 47^2=2209 8 50625 = 15 ^ 4, 50653 = 37 ^ 3, 28 225^2=50625, 226^2=51076 9 59049 = 3 ^ 10, 59319 = 39 ^ 3, 270 243^2=59049, 244^2=59536 CROSSREFS The corresponding upper members of the pairs are A340701. Cf. A001597, A076467, A097056, A340642, A340702, A340703, A340704, A340705, A340706. Cf. A117934 (excluding pairs where one of the members is a square). Sequence in context: A304557 A088248 A319389 * A106200 A303972 A361147 Adjacent sequences: A340697 A340698 A340699 * A340701 A340702 A340703 KEYWORD nonn AUTHOR Hugo Pfoertner, Jan 16 2021 STATUS approved

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Last modified May 25 15:42 EDT 2024. Contains 372800 sequences. (Running on oeis4.)