A340700 Lower of a pair of adjacent perfect powers, both with exponents > 2.

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

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