A340700 -id:A340700 - cp's OEIS frontend

A340702 Root of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

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

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

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

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

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

A340704 Exponent of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

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

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

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

A340706 Difference between upper and lower member of a pair of adjacent perfect powers A340700 and A340701, both with exponents > 2.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

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

			See A340700.

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.

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

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

A340701 Upper of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

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

The corresponding lower members of the pairs are A340700.

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

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

A340703 Root of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

2, 3, 2, 2, 2, 11, 13, 37, 39, 58, 6, 26, 28, 146, 23, 69, 294, 339, 13, 458, 508, 53, 797, 905, 927, 1077, 215, 217, 1771, 1797, 283, 358, 373, 2908, 3296, 526, 5196, 649, 691, 6334, 6502, 728, 730, 6783, 6897, 772, 811, 837, 8115, 8786, 9182, 1115, 12908, 14363

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

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

Cf. A001597, A076467, A340700, A340701, A340702, A340704, A340705, A340706.

A340701(n) = a(n)^A340705(n).

A340705 Exponent of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

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

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

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

A117934 Perfect powers (A001597) that are close, that is, between consecutive squares.

Original entry on oeis.org

27, 32, 125, 128, 2187, 2197, 6434856, 6436343, 312079600999, 312079650687, 328080401001, 328080696273, 11305786504384, 11305787424768, 62854898176000, 62854912109375, 79723529268319, 79723537443243, 4550858390629024

Offset: 1

T. D. Noe, Apr 03 2006

nonn

It appears that all pairs of close powers involve a cube. For three pairs, the other power is a 7th power. For all remaining pairs, the other power is a 5th power. If this is true, then three powers are never close.
For the first 360 terms, 176 pairs are a cube and a 5th power. The remaining four pairs are a cube and a 7th power. - Donovan Johnson, Feb 26 2011
Loxton proves that the interval [n, n+sqrt(n)] contains at most exp(40 log log n log log log n) powers for n >= 16, and hence there are at most 2*exp(40 log log n log log log n) between consecutive squares in the interval containing n. - Charles R Greathouse IV, Jun 25 2017

			27 and 32 are close because they are between 25 and 36.

Donovan Johnson, Table of n, a(n) for n = 1..360
Daniel J. Bernstein, Detecting perfect powers in essentially linear time, Mathematics of Computation 67 (1998), pp. 1253-1283.
John H. Loxton, Some problems involving powers of integers, Acta Arithmetica 46:2 (1986), pp. 113-123. See Bernstein, Corollary 19.5, for a correction to the proof of Theorem 1.
StackExchange MathOverflow, Are there ever three perfect powers between consecutive squares? Answers by Gjergji Zaimi and Felipe Voloch (2011).

Cf. A097056, A117896 (number of perfect powers between consecutive squares n^2 and (n+1)^2).

Cf. A340696, A340700.

nMax=10^14; lst={}; log2Max=Ceiling[Log[2,nMax]]; bases=Table[2,{log2Max}]; powers=bases^Range[log2Max]; powers[[1]]=Infinity; currPP=1; cnt=0; While[nextPP=Min[powers]; nextPP <= nMax, pos=Flatten[Position[powers,nextPP]]; If[MemberQ[pos,2], cnt=0, cnt++ ]; If[cnt>1, AppendTo[lst,{currPP,nextPP}]]; Do[k=pos[[i]]; bases[[k]]++; powers[[k]]=bases[[k]]^k, {i,Length[pos]}]; currPP=nextPP]; Flatten[lst]

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

Original entry on oeis.org

4, 9, 25, 225, 676, 2116, 6724, 7921, 8100, 16641, 104329, 131044, 160801, 176400, 372100, 389376, 705600, 4096576, 7306209, 7884864, 47444544, 146385801, 254817369, 373262400, 607622500, 895804900, 1121580100, 1330936324, 1536875209, 2097182025, 2258435529, 2749953600

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?

			The first terms, assuming 1 being at least a cube:
.
  n   p1  x^p1  p2  a(n)  p3  z^p3
                   =y^p2
  1  >2     1   2     4   3     8
  2   3     8   2     9   4    16
  3   4    16   2    25   3    27
  4   3   216   2   225   5   243
  5   4   625   2   676   6   729

Hugo Pfoertner, Table of n, a(n) for n = 1..181 (terms < 10^18)

Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.

a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340642(50000000)

A340643 Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 2.

Original entry on oeis.org

2, 3, 5, 15, 26, 46, 82, 89, 90, 129, 323, 362, 401, 420, 610, 624, 840, 2024, 2703, 2808, 6888, 12099, 15963, 19320, 24650, 29930, 33490, 36482, 39203, 45795, 47523, 52440, 66050, 69168, 83408, 94248, 94863, 103683, 114284, 164399, 185364, 206442, 222785, 227530, 229180

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

Within the range of the data, a(n)^2 = A340642(n), i.e., no 3 immediately consecutive perfect powers x^p1, y^p2, z^p3 with min (p1, p2, p3) > 2 are seen. Is there a counterexample?

David A. Corneth, Table of n, a(n) for n = 1..611 (first 181 terms from Hugo Pfoertner)

Cf. A000290, A001597, A025479, A076467, A097054, A111245, A153158, A340642, A340700, A340701.

a340643(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(issquare(n1)&p2>2&p0>2, print1(sqrtint(n1),", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340643(10^8)

upto(n) = {n *= n; my(v = List(), res = List([2])); for(i = 2, sqrtnint(n, 3), for(e = 3, logint(n, i), listput(v, i^e) ); ); listsort(v, 1); for(i = 1, #v - 1, if(sqrtint(v[i]) + 1 == sqrtint(v[i+1]) - issquare(v[i+1]), listput(res, sqrtint(v[i+1]-issquare(v[i+1]))); ) ); res }

More terms from David A. Corneth, Jan 14 2021

A340696 Lower of a pair of adjacent perfect powers x^p and y^q with p, q >= 4.

Original entry on oeis.org

64, 243, 279841, 62742241, 418161601, 149398068185838130176

Offset: 1

Hugo Pfoertner, Jan 21 2021

If it exists, the next term is > 10^34.

There are no more terms up to 10^36. - Jon E. Schoenfield, Jan 23 2022

			  n                   a(n)                       A340697(n)
  1                     64 =     2^6,                    81 =      3^4
  2                    243 =     3^5,                   256 =      2^8
  3                 279841 =    23^4,                279936 =      6^7
  4               62742241 =    89^4,              62748517 =     13^7
  5              418161601 =   143^4,             418195493 =     53^5
  6  149398068185838130176 = 10836^5, 149398068209479362001 = 110557^4

The corresponding upper member of the pair is A340697.

Cf. A001597, A340700, A340701.

cp's OEIS Frontend

A340702 Root of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

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A340704 Exponent of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

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A340706 Difference between upper and lower member of a pair of adjacent perfect powers A340700 and A340701, both with exponents > 2.

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A340701 Upper of a pair of adjacent perfect powers, both with exponents > 2.

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A340703 Root of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

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A340705 Exponent of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

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A117934 Perfect powers (A001597) that are close, that is, between consecutive squares.

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Mathematica

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

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PARI

A340643 Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 2.

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PARI

PARI