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CJam (2780 sequences)

{:ZA3#:Cb(40-_!!:B-\+CbB)/C),{mq_i>},=:NmQ:M;K{)[N0{N1$_*-@/M@+1$md@M@-}K*]<W%B{X0@{2$*+\}%}*ZB&=}%} 

This gives correct answers for the inclusive ranges

• [A040002, A040003]

• [A040005, A040008]

• [A040011, A040013]

• A040015

• [A040019, A040022]

• A040024

• [A040029, A040033]

• A040035

• A040037

• [A040041, A040043]

• A040048

• A040052

• [A040055, A040057]

• A040059

• A040063

• [A040071, A040074]

• A040077

• A040080

• [A040090, A040091]

• [A040093, A040094]

• A040099

• [A040109, A040111]

• A040118

• A040120

• [A040131, A040135]

• A040137

• A040139

• [A040142, A040143]

• A040151

• [A040155, A040157]

• A040166

• A040168

• [A040181, A040183]

• [A040185, A040968]

• [A041006, A041011]

• [A041014, A042937]

The A040??? sequences correspond to the continued fractions of non-rational square roots from sqrt(5) to sqrt(1000) (with the gaps corresponding to ones which appear earlier in OEIS, but conveniently filled with random sequences); the A041??? sequences correspond to the numerators and denominators of the continued fraction convergents for non-rational square roots from sqrt(6) to sqrt(1000) (with the gap corresponding to sqrt(10), at A005667 and A005668).

The answer ports elements of two earlier answers of mine in GolfScript:

• Determining the continued fractions of square roots
• Good rational approximations of pi

It was tricky to map the sequence number to the value to sqrt. In the end I couldn't find anything better than generating the list of non-squares and selecting by index. The trick I used in an earlier version to have an all-zero fallback for out-of-range indices no longer fits: it took me a couple of hours just to get both continued fractions and convergents into the 100 byte limit.