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A001597 Perfect powers: m^k where m > 0 and k >= 2.
(Formerly M3326 N1336)
249
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A175064 for number of ways to write a(n) as m^k (m >= 1, k >= 1). - Jaroslav Krizek, Jan 23 2010

a(1) = 1, for n >= 2: a(n) = numbers m such that sum of perfect divisors of x = m has no solution. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) for n >= 2 is complement of A175082. - Jaroslav Krizek, Jan 24 2010

A075802(a(n)) = 1. - Reinhard Zumkeller, Jun 20 2011

Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.

For a proof of Catalan's conjecture, see the paper by Metsänkylä. - L. Edson Jeffery, Nov 29 2013

m^k is the largest number n such that (n^k-m)/(n-m) is an integer (for k > 1 and m > 1). - Derek Orr, May 22 2014

a(n) is asymptotic to n^2, since the density of cubes and higher powers among the squares and higher powers is 0. E.g.

  a(10^1) = 49 (49% of 10^2),

  a(10^2) = 6400 (64% of 10^4),

  a(10^3) = 804357 (80.4% of 10^6),

  a(10^4) = 90706576 (90.7% of 10^8),

  a(10^n) ~ 10^(2n) - o(10^(2n)). - Daniel Forgues, Jul 22 2014

A proper subset of A001694. - Robert G. Wilson v, Aug 11 2014

a(10^n): 1, 49, 6400, 804357, 90706576, 9565035601, 979846576384, 99066667994176, 9956760243243489, …, . - Robert G. Wilson v, Aug 15 2014

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.

R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, p. 1.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

David W. Wilson, Table of n, a(n) for n = 1..10000.

Abdelkader Dendane, Power (Exponential) Calculator.

Serhat Sevki Dincer, Powers up to 2^50.

H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), p. 268, problem H-170.

Rafael Jakimczuk, Asymptotic Formulae for the n-th Perfect Power, Journal of Integer Sequences, Vol. 15 (2012), #12.5.5.

Tauno Metsänkylä, Catalan's conjecture: another old Diophantine problem solved, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.

Donald J. Newman, A Problem Seminar, Springer; see Problem #72.

Alf van der Poorten, Remarks on the sequence of 'perfect' powers.

Michel Waldschmidt, Open Diophantine problems, arXiv:math.NT/0312440.

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.

Eric W. Weisstein, MathWorld: Perfect Power

FORMULA

Goldbach showed that Sum_{n >= 2} 1/(a(n)-1) = 1.

Formulas from postings to the Number Theory List by various authors, 2002:

Sum_{i = 2}^{infty} sum_{j = 2}^{infty} 1/i^j = 1;

Sum_{k = 2}^infty 1/(a(k)+1) = pi^2 / 3 - 5/2;

Sum_{k = 2}^infty 1/a(k) = sum_{n = 2}^infinity mu(n)(1- zeta(n)) approx = .87446436840494... See A072102.

For asymptotics see Newman.

For n > 1: GCD(exponents in prime factorization of a(n)) > 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012

a(n) ~ n^2. - Thomas Ordowski, Nov 04 2012

MAPLE

isA001597 := proc(n) local e ; e := seq(op(2, p), p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:

A001597 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001597(a) then return a ; end if; end do; end if; end proc:

seq(A001597(n), n=1..70) ; # R. J. Mathar, Jun 07 2011

MATHEMATICA

min = 0; max = 10^4;  Union@ Flatten@ Table[ n^expo, {expo, Prime@ Range@ PrimePi@ Log2@ max}, {n, Floor[1 + min^(1/expo)], max^(1/expo)}] (* T. D. Noe, Apr 18 2011 and slightly modified by Robert G. Wilson v, Aug 11 2014 *)

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1765, perfectPowerQ] (* Ant King, Jun 29 2013 and slightly modified by Robert G. Wilson v, Aug 11 2014 *)

nextPerfectPower[n_] := If[n == 1, 4, Min@ Table[ (Floor[n^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ n}]]; NestList[ nextPerfectPower, 1, 55] (* Robert G. Wilson v, Aug 11 2014 *)

PROG

(MAGMA) [1] cat [n : n in [2..1000] | IsPower(n) ];

(PARI) {a(n) = local(m, c); if( n<2, n==1, c=1; m=1; while( c<n, m++; if( ispower(m), c++)); m)} /* Michael Somos, Aug 05 2009 */

(Sage)

def A001597_list(n) :

    return [k for k in (1..n) if k.is_perfect_power()]

A001597_list(1764) # Peter Luschny, Feb 03 2012

(Haskell)

import Data.Map (singleton, findMin, deleteMin, insert)

a001597 n = a001597_list !! (n-1)

(a001597_list, a025478_list, a025479_list) =

   unzip3 $ (1, 1, 2) : f 9 (3, 2) (singleton 4 (2, 2)) where

   f zz (bz, ez) m

    | xx < zz = (xx, bx, ex) :

                f zz (bz, ez+1) (insert (bx*xx) (bx, ex+1) $ deleteMin m)

    | xx > zz = (zz, bz, 2) :

                f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)

    | otherwise = f (zz+2*bz+1) (bz+1, 2) m

    where (xx, (bx, ex)) = findMin m  --  bx ^ ex == xx

-- Reinhard Zumkeller, Mar 28 2014, Oct 04 2012, Apr 13 2012

CROSSREFS

Cf. A023055, A023057, A025478, A070428, A072102, A074981.

Cf. A089579, A089580 (number of exact powers < 10^n).

Complement of A007916.

Cf. A076404; subsequence of A072103; union of A075109 and A075090.

Cf. A239728, A239870, A097054.

Cf. A072777 (subsequence).

Sequence in context: A080366 A001694 A157985 * A072777 A076292 A090516

Adjacent sequences:  A001594 A001595 A001596 * A001598 A001599 A001600

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

Minor corrections from N. J. A. Sloane, Jun 27 2010

STATUS

approved

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Last modified September 2 12:55 EDT 2014. Contains 246357 sequences.