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A001597
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Perfect powers: m^k where m > 0 and k >= 2.
(Formerly M3326 N1336)
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230
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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)
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OFFSET
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1,2
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COMMENTS
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Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.
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]
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REFERENCES
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H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), 268.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
D. J. Newman, A Problem Seminar, Springer; see Problem #72.
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).
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LINKS
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David W. Wilson, Table of n, a(n) for n = 1..10000
A. Dendane, Power (Exponential) Calculator
Serhat Sevki Dincer, Powers up to 2^50
Alf van der Poorten, Remarks on the sequence of 'perfect' powers
M. Waldschmidt, Open Diophantine problems, arXiv:math.NT/0312440
Eric Weisstein's World of Mathematics, Perfect Power
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FORMULA
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Goldbach showed that Sum_{n >= 2} 1/(a(n)-1) = 1.
Formulae 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}^infty mu(n)(1- zeta(n)) approx = .87446436840494...
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
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MAPLE
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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
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MATHEMATICA
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nn = 10^4; Join[{1}, Union[Flatten[Table[n^i, {i, Prime[Range[PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]]] (* T. D. Noe, Apr 18 2011 *)
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PROG
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(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 = 1 : f 9 3 (Map.singleton 4 2) where
f zz z m
| xx < zz = xx : f zz z (Map.insert (x * xx) x $ Map.deleteMin m)
| xx > zz = zz : f (zz + 2 * z + 1) (z + 1) (Map.insert (z * zz) z m)
| otherwise = f (zz + 2 * z + 1) (z + 1) m
where (xx, x) = Map.findMin m
-- Reinhard Zumkeller, Oct 04 2012, Apr 13 2012
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CROSSREFS
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Cf. A023055, A023057, A070428, A074981, A025478.
Cf. A089579, A089580 (number of exact powers < 10^n).
Complement of A007916.
Cf. A076404; subsequence of A072103; union of A075109 and A075090.
Sequence in context: A080366 A001694 A157985 * A072777 A076292 A090516
Adjacent sequences: A001594 A001595 A001596 * A001598 A001599 A001600
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Minor corrections from N. J. A. Sloane, Jun 27 2010
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STATUS
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approved
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