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OEIS sequence A052349 is given by:

Lexicographically earliest sequence such that no subset sums to a prime.

1, 8, 24, 25, 86, 1260, 1890, 14136, 197400, 10467660, 1231572090, 682616834970

I don't understand why the 4th term isn't also 24. It seems that $[1,8,24,24]$ contains no subsets such that the sum is prime.

It's possible that the sequence is simply incorrect, but I want to get a second pair of eyes before suggesting that edit.

Edit: Can anything be said of the asymptotic behavior of this sequence?

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All subsets: $$ \begin{alignat}{2} &1 + 24 + 24 &&= 49 \\ &1 + 24 &&= 25 \\ &1 + 8 + 24 + 24 &&= 57 \\ &1 + 8 + 24 &&= 33 \\ &1 + 8 &&= 9 \\ &1 &&= 1 \\ &24 + 24 &&= 48 \\ &24 &&= 24 \\ &8 + 24 + 24 &&= 56 \\ &8 + 24 &&= 32 \\ &8 &&= 8 \\ &\text{(empty sum)} &&= 0 \end{alignat} $$

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    Twenty "five" comes before twenty "four" lexicographically (i < o, the second letters of five and four), I think that is the reason. – 2017-01-07
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    Maybe they meant that there couldn't be repeats, but failed to put that in the definition. Also don't forget, if you change that, it changes the rest of the sequence. – 2017-01-07
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    @астонвіллаолофмэллбэрг I don't think that's it, because "eight" would then come before "one". – 2017-01-07
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    @setholopolus That's my suspicion too. – 2017-01-07
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    @PeterKagey Ah, that's right! Sorry for the confusion. Yes, it looks like the sequence wants to avoid repeats. But then, where does "lexicographically" come in? Or is the word incorrectly used? – 2017-01-07
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    @астонвіллаолофмэллбэрг ["Lexicographic"](https://en.wikipedia.org/wiki/Lexicographical_order) would be an ordering on the infinite sequences such that $(a_1,a_2,a_3,\ldots ) \prec (b_1,b_2,b_2,\ldots)$ iff there is an $n$ such that $a_i = b_i$ for all $i < n$ and $a_n < b_n$. Note that $25,24,8,1,\ldots$ is the start of another sequence of the same sort, but it comes lexicographically after. – 2017-01-07

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With the help from a comment by setholopolus along with feedback from OEIS editors Omar E. Pol and N. J. A. Sloane, the sequence name of A052349 was changed to

Lexicographically earliest sequence of distinct positive integers such that no subsequence sums to a prime.

and sequence A280708 was added with the "distinct" condition relaxed.

Something curious: only a few terms of each sequence is known, but the sequences appear to differ only at the fourth term.

A052349: 1, 8, 24, 25, 86, 1260, 1890, 14136, 197400, 10467660, 1231572090, 682616834970
A280708: 1, 8, 24, 24, 86, 1260, 1890, 14136, 197400, 10467660, 1231572090