%I A001597 M3326 N1336
%S A001597 1,4,8,9,16,25,27,32,36,49,64,81,100,121,125,128,144,169,196,216,225,243,
%T A001597 256,289,324,343,361,400,441,484,512,529,576,625,676,729,784,841,900,961,
%U A001597 1000,1024,1089,1156,1225,1296,1331,1369,1444,1521,1600,1681,1728,1764
%N A001597 Perfect powers: m^k where m is an integer and k >= 2.
%C A001597 Catalan's conjecture (now a theorem) is that 1 occurs just once as a 
               difference, between 8 and 9.
%C A001597 Goldbach showed that Sum 1/(a(n)-1) = 1.
%D A001597 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001597 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001597 H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), 268.
%D A001597 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 66.
%D A001597 D. J. Newman, A Problem Seminar, Springer; see Problem #72.
%D A001597 R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, p. 1.
%H A001597 David W. Wilson, <a href="b001597.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A001597 A. Dendane, <a href="http://www.analyzemath.com/Calculators_2/power_calculator.html">
               Power (Exponential) Calculator</a>
%H A001597 Serhat Sevki Dincer, <a href="http://jugosoft.net/math/powers_2_50.7z">
               Powers up to 2^50</a>
%H A001597 Alf van der Poorten, <a href="a023057.txt">Remarks on the sequence of 
               'perfect' powers</a>
%H A001597 M. Waldschmidt, <a href="http://arXiv.org/abs/math.NT/0312440">Open Diophantine 
               problems</a>
%H A001597 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PerfectPower.html">Perfect Power</a>
%F A001597 Formulae from postings to the Number Theory List by various authors, 
               2002:
%F A001597 Sum_{i=2}^{infty} sum_{j=2}^{infty} 1/i^j =1;
%F A001597 Sum_{k=1}^infty 1/(a_k-1)=1;
%F A001597 Sum_{k=1}^infty 1/(a_k+1)= pi^2 / 3 - 5/2;
%F A001597 Sum_{k=1}^infty 1/a_k = sum_{n=2}^infty mu(n)(1- zeta(n)) approx = .87446436840494...
%F A001597 For asymptotics see Newman.
%e A001597 x + 4*x^2 + 8*x^3 + 9*x^4 + 16*x^5 + 25*x^6 + 27*x^7 + 32*x^8 + 36*x^9 
               + ...
%t A001597 Union[ Join[{1}, Flatten[ Table[ n^i, {n, 2, Sqrt[1800]}, {i, 2, Log[n, 
               1800]}]]]]
%t A001597 Join[{1}, Select[Range@1848, GCD @@ Last /@ FactorInteger@# > 1 &]] (* 
               or *)
%o A001597 (MAGMA) [1] cat [n : n in [2..1000] | IsPower(n) ];
%o A001597 (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 */
%Y A001597 Cf. A023055, A023057, A070428, A074981, A025478.
%Y A001597 Cf. A089579, A089580 (number of exact powers < 10^n).
%Y A001597 Complement of A007916.
%Y A001597 Sequence in context: A080366 A001694 A157985 this_sequence A072777 A076292 
               A090516
%Y A001597 Adjacent sequences: A001594 A001595 A001596 this_sequence A001598 A001599 
               A001600
%K A001597 nonn,easy,nice
%O A001597 1,2
%A A001597 N. J. A. Sloane (njas(AT)research.att.com).