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( binomial(m,k)/(k+1)^2, k=0..m) (binomial transform of 1/(k+1)^2), has the same numerator than g(m) = sum( (2^(k+1)-1)/(k+1), k=0..m) (which are also the partial sums of the binomial transformation of 1/(k+1)).
Obviously, f(m) = sum( binomial(m+1,k+1)/(k+1), k=0..m)/(m+1) and since g(m) = (m+1) f(m) (cf. notes by R. J. Mathar on A097345), g(m) = sum( binomial(m+1,k)/k, k=1..m+1).
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 co-prime 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(M-1,k-1)/k^2)) is not a multiple of M = numbers M such that numerator( sum( (2^k-1)/k, k=1..M)) is not co-prime to M.