The terms 59 and 1519 were found by Daniel Glasscock (glasscock(AT)rice.edu), Jan 04 2008.
a(6) > 10^5.
These are the numbers m such that f(m) = Sum_{k=0..m} binomial(m,k)/(k+1)^2 (binomial transform of 1/(k+1)^2) has the same numerator as g(m) = Sum_{k=0..m} (2^(k+1)  1)/(k+1) (which are also the partial sums of the binomial transformation of 1/(k+1)).
Obviously, f(m) = Sum_{k=0..m} binomial(m+1, k+1)/((k+1)*(m+1)) and since g(m) = (m+1) f(m) (cf. notes by R. J. Mathar on A097345), g(m) = Sum_{k=1..m+1} binomial(m+1,k)/k.
We have the equivalences: numerator(g(n)) = numerator(f(n)) <=> (n+1)  denominator(f(n)) <=> gcd(numerator(g(n)), n+1) = 1.
Therefore this sequence can be alternatively defined in either of the following two ways: numbers n such that the denominator of f(n) is not divisible by (n+1); numbers n such that the numerator of g(n) is not coprime to (n+1).
In terms of M = m+1, the characterization reads: a(n)+1 = numbers M such that denominator(Sum_{k=1..M} binomial(M1, k1)/k^2) is not a multiple of M = numbers M such that numerator(Sum_{k=1..M} (2^k  1)/k) is not coprime to M.
