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 A001597 Perfect powers: m^k where m > 0 and k >= 2. (Formerly M3326 N1336) 450
 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 Might also be called the nontrivial powers. - N. J. A. Sloane, Mar 24 2018 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 From Daniel Forgues, Jul 22 2014: (Start) 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)). (End) 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. H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), p. 268, problem H-170. R. Jakimczuk, On the Distribution of Perfect Powers, J. Int. Seq. 14 (2011) # 11.8.5 Rafael Jakimczuk, Asymptotic Formulae for the n-th Perfect Power, Journal of Integer Sequences, Vol. 15 (2012), #12.5.5. Holly Krieger and Brady Haran, Catalan's Conjecture, Numberphile video (2018) 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. M. A. Nyblom, A Counting Function for the Sequence of Perfect Powers, Austral. Math. Soc. Gaz. 33 (2006), 338-343. Alf van der Poorten, Remarks on the sequence of 'perfect' powers. Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004. 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} Sum_{j >= 2} 1/i^j = 1; Sum_{k >= 2} 1/(a(k)+1) = Pi^2 / 3 - 5/2; Sum_{k >= 2} 1/a(k) = Sum_{n >= 2} mu(n)(1- zeta(n)) approx = 0.87446436840494... 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)) ;     return ( igcd(e) >=2 or n =1 ) ; 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; 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; 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 *) Join[{1}, Select[Range, GCD@@FactorInteger[#][[All, 2]]>1&]] (* Harvey P. Dale, Apr 30 2018 *) PROG (MAGMA)  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 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 Complement of A007916. Subsequence of A072103; A072777 is a subsequence. Union of A075109 and A075090. Cf. A023055, A023057, A025478, A070428, A072102, A074981, A076404, A239728, A239870, A097054, A089579, A089580 (number of exact powers < 10^n). There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2), and which are sometimes confused with the present sequence. First differences give A053289. Sequence in context: A001694 A317102 A157985 * A072777 A076292 A090516 Adjacent sequences:  A001594 A001595 A001596 * A001598 A001599 A001600 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Minor corrections from N. J. A. Sloane, Jun 27 2010 STATUS approved

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)