1,2

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.

See A175064 for number of ways to write a(n) as m^k (m >= 1, k >= 1). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 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 A157082. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010]

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

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

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.

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

Eric Weisstein's World of Mathematics, Perfect Power

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.

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 + ...

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 *)

(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 */

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

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

Complement of A007916.

Sequence in context: A080366 A001694 A157985 this_sequence A072777 A076292 A090516

Adjacent sequences: A001594 A001595 A001596 this_sequence A001598 A001599 A001600

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).