A117934 -id:A117934 - 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).

A097056 Numbers n such that the interval n^2 < x < (n+1)^2 contains two or more distinct nonsquare perfect powers A097054.

Original entry on oeis.org

5, 11, 46, 2536, 558640, 572783, 3362407, 7928108, 8928803, 67460050, 106938971, 1763350849, 2501641555, 2756149047, 4584349318, 5713606932, 17941228664, 375376083513, 411124334926, 452894760105, 1167680330892, 1933159894790, 1946131548918, 2506032014606, 2507269866902, 8217688694093

Offset: 1

Hugo Pfoertner, Jul 21 2004

nonn

Empirically, there seem to be no intervals between consecutive squares containing more than two nonsquare perfect powers.

It is easy to see that two distinct powers between n^2 and (n+1)^2 are necessarily of the form x^p and y^q where p, q are distinct odd primes. Among the first 180 terms, only 4 are of type (p,q) = (3,7) and all others are of type (3,5). The first term with q = 11, if it exists, is > (1e6)^(11/2) = 1e33. - M. F. Hasler, Jan 18 2021

			a(1) = 5: 5^2 < 3^3 < 2^5 < 6^2,
a(2) = 11: 11^2 < 5^3 < 2^7 < 12^2,
a(3) = 46: 46^2 = 2116 < 3^7 = 2187 < 13^3 = 2197 < 47^2 = 2209.
a(4) = 2536: 2536^2 = 6431296 < 186^3 = 6434856 < 23^5 = 6436343 < 2537^2 = 6436369.
22 is not in the sequence because 2^9 and 8^3 (22^2 < 512 < 23^2) are not distinct.
Also, 181 is not listed since between 181^2 and 182^2 there is only 32^3 = 8^5.

T. D. Noe, Table of n, a(n) for n = 1..180 (using the b-file from A117934)

Cf. A000290, A097054, A097055.

Cf. A173341 (q=5), A173342 (q=7): y with a(n)^2 < y^q < (a(n)+1)^2.

is(n)=my(s,t); forprime(p=3,2*log(n+1.5)\log(2), t=floor((n+1)^(2/p)); if(t^p>n^2 && !ispower(t) && s++ > 1, return(1))); 0 \\ Charles R Greathouse IV, Dec 11 2012

haspow(lower,upper,eMin,eMax)=if(sqrtnint(upper,3)^3>lower, return(1)); forprime(e=eMin,eMax, if(sqrtnint(upper,e)^e>lower, return(1))); 0
list(lim)=lim\=1; my(v=List(),M=(lim+1)^2,L=logint(M,2),s); forprime(e=5,L, forprime(p=2,sqrtnint(M,e), s=sqrtint(p^e); if(haspow(s^2,(s+1)^2-1,e+1,L) && s<=lim, listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Nov 05 2015

a(5)-a(20) from Don Reble

a(21)-a(26) from David Wasserman, Dec 17 2007

A117896 Number of perfect powers between consecutive squares n^2 and (n+1)^2.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Mar 31 2006, Feb 15 2010

nonn

a(n)=2 only 14 times for n^2 < 2^63. What is the least n such that a(n)=3? Is a(n) bounded?

			a(5)=2 because powers 27 and 32 are between 25 and 36.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. H. Loxton, Some problems involving powers of integers, Acta Arith., 46 (1986), pp. 113-123.
C. L. Stewart, On heights of multiplicatively dependent algebraic numbers, Acta Arith. 133 (2008), pp. 97-108.
J. Turk, Multiplicative properties of integers in short intervals, Proc. Kon. Ned. Akad. Wet. (A) 83 (1980), pp. 429-436.

Cf. A001597 (perfect powers), A014085 (primes between squares), A097055, A097056, A117934.

nn=151^2; powers=Join[{1}, Union[Flatten[Table[n^i, {i,Prime[Range[PrimePi[Log[2,nn]]]]}, {n,2,nn^(1/i)}]]]]; t=Table[0,{Sqrt[nn]-1}]; Do[n=Floor[Sqrt[i]]; If[i>n^2, t[[n]]++], {i,powers}]; t (* revised, T. D. Noe, Apr 19 2011 *)

a(n)=my(k);-sum(e=3,2*log(n+1)\log(2),k=round((n+1/2)^(2/e))^e;if(n^2Charles R Greathouse IV, Dec 19 2011

Trivially, a(n) << log n/log log n. Turk gives a(n) << sqrt(log n) and Loxton improves this to a(n) <= exp(40 sqrt(log log n log log log n)). Stewart improves the constant from 40 to 30 and conjectures that a(n) < 3 for all but finitely many n. - Charles R Greathouse IV, Dec 11 2012

A117594 Numbers whose fifth powers are closer to cubic numbers than square numbers.

Original entry on oeis.org

199, 1354, 4995, 7320, 7994, 12634, 44217, 91116, 177682, 394826, 458908, 462763, 512012, 1706886, 1738064, 1801677, 1880465, 2523441, 5691648, 6714911, 8383950, 8403388, 11100341, 14706104, 14706146, 15460136, 16337238, 18898872, 21194961

Offset: 1

Ed Pegg Jr, Apr 05 2006

nonn

Numbers which are cubes themselves are excluded as trivial.

It appears that this sequence is infinite. For seventh powers < 10^49, only 2^7 and 3^7 are closer to cubes than squares. Note that 1/2+1/3+1/5>1, but 1/2+1/3+1/7<1. Do these inequalities determine whether there are an infinite or finite number of solutions? Mazur discusses how the ABC conjecture applies to perfect power problems. - T. D. Noe, Apr 07 2006

			The distance of 199^5 to the nearest cube is 49688. To the nearest square is 165882.

B. Mazur, Questions about Number
Eric Weisstein's World of Mathematics, MathWorld: Perfect Power

Cf. A117934 (perfect powers that are close).

nMax=10^5; lst={}; Do[n5=n^5; n3=Round[n5^(1/3)]^3; n2=Round[n5^(1/2)]^2; If[0 < Abs[n5-n3] < Abs[n5-n2], AppendTo[lst,n]], {n,nMax}]; lst (* T. D. Noe, Apr 07 2006 *)

More terms from T. D. Noe and Hans Havermann, Apr 08 2006

cp's OEIS Frontend

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

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A097056 Numbers n such that the interval n^2 < x < (n+1)^2 contains two or more distinct nonsquare perfect powers A097054.

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A117896 Number of perfect powers between consecutive squares n^2 and (n+1)^2.

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PARI

Formula

A117594 Numbers whose fifth powers are closer to cubic numbers than square numbers.

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Keywords

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Programs

Mathematica

Extensions