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Search: id:A001597
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| A001597 |
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Perfect powers: m^k where m is an integer and k >= 2.
(Formerly M3326 N1336)
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+0
176
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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; listen)
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OFFSET
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1,2
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COMMENT
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Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.
Goldbach showed that Sum 1/(a(n)-1) = 1.
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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.
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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, Table up to 10^9
Alf van der Poorten, Remarks on the sequence of 'perfect' powers
M. Waldschmidt, Open Diophantine problems
Eric Weisstein's World of Mathematics, Perfect Power
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FORMULA
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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=1}^infty 1/(a_k-1)=1;
Sum_{k=1}^infty 1/(a_k+1)= pi^2 / 3 - 5/2;
Sum_{k=1}^infty 1/a_k = sum_{n=2}^infty mu(n)(1- zeta(n)) approx = .87446436840494...
For asymptotics see Newman.
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MATHEMATICA
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Union[ Join[{1}, Flatten[ Table[ n^i, {n, 2, Sqrt[1800]}, {i, 2, Log[n, 1800]}]]]]
Join[{1}, Select[Range@1848, GCD @@ Last /@ FactorInteger@# > 1 &]] (* or *)
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PROGRAM
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(MAGMA) [1] cat [n : n in [2..1000] | IsPower(n) ];
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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.
Adjacent sequences: A001594 A001595 A001596 this_sequence A001598 A001599 A001600
Sequence in context: A109422 A080366 A001694 this_sequence A072777 A076292 A090516
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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