A340642 -id:A340642 - cp's OEIS frontend

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

Hugo Pfoertner, Jan 16 2021

nonn

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.

			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

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.

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).

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

A342561 List points (x,y,z) having integer coordinates, sorted first by R^2 = x^2 + y^2 + z^2 and in case of ties, then by z and last by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

Original entry on oeis.org

0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 1, -1, -1, 1, 1, 0, -1, 0, 1, -1, -1, 1, 1, -1, -1, 1, 0, 2, 0, -2, 0, 0, 1, 0, -1, 0, 2, 0, -2, 0, 2, 1, -1, -2, -2, -1, 1, 2, 2, 0, -2, 0, 1, 0, -1, 0, 1, -1, -1, 1, 2, 1, -1, -2, -2, -1, 1, 2, 2, 1, -1, -2, -2, -1, 1, 2, 1, -1, -1, 1, 2, 0, -2, 0, 2, -2, -2, 2, 2, 0, -2, 0

Offset: 0

Hugo Pfoertner, Apr 27 2021

sign

This is a 3-dimensional generalization of A305575 and A305576.
y-coordinates are in A342562, z-coordinates are in A342563.
These lists can be read as an irregular table, where row r lists the respective coordinates of the points on the sphere with radius R = sqrt(r); their number (i.e., the row length) is given by A005875 = (1, 6, 12, 8, 6, 24, 24, 0, 12, 30, ...). - M. F. Hasler, Apr 27 2021

			   n    x    y    z  R^2  phi/Pi
   0    0    0    0   0   0.000
   1    0    0   -1   1   0.000
   2    1    0    0   1   0.000
   3    0    1    0   1   0.500
   4   -1    0    0   1   1.000
   5    0   -1    0   1   1.500
   6    0    0    1   1   0.000
   7    1    0   -1   2   0.000
   8    0    1   -1   2   0.500
   9   -1    0   -1   2   1.000
  10    0   -1   -1   2   1.500
  11    1    1    0   2   0.250
  12   -1    1    0   2   0.750
  13   -1   -1    0   2   1.250
  14    1   -1    0   2   1.750
  15    1    0    1   2   0.000
  16    0    1    1   2   0.500
  17   -1    0    1   2   1.000
  18    0   -1    1   2   1.500
  19    1    1   -1   3   0.250
  20   -1    1   -1   3   0.750
  21   -1   -1   -1   3   1.250
  22    1   -1   -1   3   1.750
  23    1    1    1   3   0.250
  24   -1    1    1   3   0.750
  25   -1   -1    1   3   1.250
  26    1   -1    1   3   1.750
  27    0    0   -2   4   0.000
  28    2    0    0   4   0.000
  29    0    2    0   4   0.500

Hugo Pfoertner, Table of n, a(n) for n = 0..10130

Cf. A005875, A117609, A305575, A305576, A342562, A342563.

Cf. A343630, A340631, A340632, A343633 for a variant which "connects" corresponding poles of successive shells, A343640, A340641, A340642, A343643 for a square spiral variant.

shell(n, Q=Qfb(1,0,1), L=List())={for(z=if(n, sqrtint((n-1)\3)+1), sqrtint(n), my(S=if(n>z^2, Set(apply(vecsort, abs(qfbsolve(Q, n-z^2, 3)))), [[0,0]])); foreach(S, s, forperm(concat(s,z), p, listput(L, p)))); for(i=1,3, for(j=1,#L, my(X=L[j]); (X[i]*=-1) && listput(L,X))); vecsort(L, (p,q)->if( p[3]!=q[3], p[3]-q[3], p[1]==q[1], q[2]-p[2], p[2]*q[2]<0, q[2]-p[2], (q[1]-p[1])*(p[2]+q[2])))} \\ Gives list of all points with Euclidean norm sqrt(n).
A342561_vec=concat([[P[1] | P <- shell(n)] | n<-[0..7]]) \\ M. F. Hasler, Apr 27 2021

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

cp's OEIS Frontend

A340700 Lower of a pair of adjacent perfect powers, 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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A342561 List points (x,y,z) having integer coordinates, sorted first by R^2 = x^2 + y^2 + z^2 and in case of ties, then by z and last by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

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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

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