cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Views

Author

Hugo Pfoertner, Jan 16 2021

Keywords

Examples

			See A340700.

Links

Crossrefs

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

Formula

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

Views

Author

Hugo Pfoertner, Jan 16 2021

Keywords

Examples

			See A340700.

Links

Crossrefs

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

Formula

A340701(n) = A340703(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

Views

Author

Hugo Pfoertner, Jan 16 2021

Keywords

Comments

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

Examples

			See A340700.

Links

Crossrefs

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

Formula

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

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

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, Jan 16 2021

Keywords

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.

Examples

			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

Links

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).

Formula

a(n) = A340702(n)^A340704(n) = A340701(n) - A340706(n).

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

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, Jan 16 2021

Keywords

Examples

			See A340700.

Links

Crossrefs

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

Formula

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

Views

Author

Hugo Pfoertner, Jan 16 2021

Keywords

Examples

			See A340700.

Links

Crossrefs

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

Formula

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

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

Views

Author

Hugo Pfoertner, Jan 14 2021

Keywords

Comments

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?

Examples

			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

Links

Crossrefs

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

Programs

  • PARI
    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

Views

Author

Hugo Pfoertner, Jan 14 2021

Keywords

Comments

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?

Links

Crossrefs

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

Programs

  • PARI
    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)
  • PARI
    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 }

Extensions

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

Views

Author

Hugo Pfoertner, Jan 21 2021

Keywords

Comments

If it exists, the next term is > 10^34.
There are no more terms up to 10^36. - Jon E. Schoenfield, Jan 23 2022

Examples

			  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

Crossrefs

The corresponding upper member of the pair is A340697.
Cf. A001597, A340700, A340701.

A340697 Upper of a pair of adjacent perfect powers x^p and y^q with p, q >= 4.

Original entry on oeis.org

81, 256, 279936, 62748517, 418195493, 149398068209479362001
Offset: 1

Views

Author

Hugo Pfoertner, Jan 21 2021

Keywords

Examples

			See A340696.

Crossrefs

The corresponding lower member of the pair is A340696.
Cf. A001597, A340700, A340701.
Showing 1-10 of 10 results.

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