A340703 -id:A340703

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A340700

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

+10
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).

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Jan 16 2021

STATUS

approved

A340701

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

+10
11

32, 81, 128, 256, 1024, 1331, 2197, 50653, 59319, 195112, 279936, 456976, 614656, 3112136, 6436343, 22667121, 25412184, 38958219, 62748517, 96071912, 131096512, 418195493, 506261573, 741217625, 796597983, 1249243533, 2136750625, 2217373921, 5554637011, 5802888573

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

FORMULA

a(n) = A340703(n)^A340705(n) = A340700(n) + A340706(n).

EXAMPLE

See A340700.

CROSSREFS

The corresponding lower members of the pairs are A340700.

Cf. A001597, A076467, A340642, A340702, A340703, A340704, A340705, A340706.

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Jan 16 2021

STATUS

approved

A340702

Root of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

+10
7

3, 2, 5, 3, 10, 6, 3, 15, 3, 21, 23, 77, 85, 42, 186, 283, 71, 79, 89, 99, 107, 143, 150, 165, 168, 188, 1288, 1304, 273, 276, 1858, 2542, 2685, 396, 435, 4246, 612, 5619, 6109, 710, 2, 6549, 6573, 199, 201, 7082, 7563, 7888, 855, 7, 938, 11562, 1211, 1312, 1438

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

FORMULA

A340700(n) = a(n)^A340704(n).

EXAMPLE

See A340700.

CROSSREFS

Cf. A001597, A076467, A340700, A340701, A340703, A340704, A340705, A340706.

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Jan 16 2021

STATUS

approved

A340704

Exponent of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

+10
7

3, 6, 3, 5, 3, 4, 7, 4, 10, 4, 4, 3, 3, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 3, 3, 3, 4, 4, 3, 4, 3, 3, 4, 38, 3, 3, 5, 5, 3, 3, 3, 4, 14, 4, 3, 4, 4, 4, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

FORMULA

A340700(n) = A340702(n)^a(n).

EXAMPLE

See A340700.

CROSSREFS

Cf. A001597, A076467, A340700, A340701, A340702, A340703, A340705, A340706.

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Jan 16 2021

STATUS

approved

A340705

Exponent of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

+10
7

5, 4, 7, 8, 10, 3, 3, 3, 3, 3, 7, 4, 4, 3, 5, 4, 3, 3, 7, 3, 3, 5, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 5, 3, 4, 4, 3, 4, 4, 4, 4, 4, 5, 4, 5, 4, 3, 4, 4, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

FORMULA

A340701(n) = A340703(n)^a(n).

EXAMPLE

See A340700.

CROSSREFS

Cf. A001597, A076467, A340700, A340701, A340702, A340703, A340704, A340706.

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Jan 16 2021

STATUS

approved

A340706

Difference between upper and lower member of a pair of adjacent perfect powers A340700 and A340701, both with exponents > 2.

+10
7

5, 17, 3, 13, 24, 35, 10, 28, 270, 631, 95, 443, 531, 440, 1487, 1934, 503, 8138, 6276, 12311, 16911, 33892, 11573, 17000, 3807, 45197, 30753, 31457, 65170, 105597, 127209, 206808, 109516, 139456, 377711, 530040, 561600, 690742, 952332, 457704, 671064, 353107

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The differences are expected to be bounded below by the Lang-Waldschmidt conjecture (see Waldschmidt 2013, p. 6, Conjecture 6).

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

FORMULA

a(n) = A340701(n) - A340700(n).

EXAMPLE

See A340700.

CROSSREFS

Cf. A001597, A076467, A340700, A340701, A340702, A340703, A340704, A340705.

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Jan 16 2021

STATUS

approved

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