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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
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. A117934 (excluding pairs where one of the members is a square).
Sequence in context: A304557 A088248 A319389 * A106200 A303972 A361147
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.)