17
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Here are some numbers:
959, 233, 14, 36, 105, 45, 784, 120, 1579, 1001, 1111, 90, 841, 1607, 590, 625, 609, 144, 606
And here are the rest of the numbers:
1, 2, 5, 20, 30, 34, 42, 55, 56, 66, 72, 78, 89, 91, 132, 136, 153, 377, 429, 512, 595, 598, 602, 610, 676, 729, 900, 949, 969, 979, 987, 989, 999, 1000, 1221, 1331, 1583, 1597, 1601

What do I hope that this puzzle can be for other number sequence puzzles?

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These numbers are all

part of multiple different sequences. For instance, we have 36, 45, 55, 66, 78, 91, 105, 120, 136, and 153: all triangular numbers.

This might be hard to get a foothold on normally, but

examining the post (by clicking "edit") shows us that an image has been uploaded but left out of the post.

enter image description here

This image helpfully gives us the lengths of all the sequences, and where they intersect! (For instance, the Fibonacci numbers are the top most long bar, and they share a number with three other sequences.)

So we can find all the partial sequences:

Fibonacci numbers: 34, 55, 89, 144, 233, 377, 610, 987, 1597
Triangular numbers: 36, 45, 66, 78, 91, 105, 120, 136, 153
Sphenic numbers: 590, 595, 598, 602, 606, 609, 610
Prime numbers: 1579, 1583, 1597, 1601, 1607
Catalan numbers: 1, 2, 5, 14, 42, 132, 429
Oblong numbers: 20, 30, 42, 56, 71, 90
Square numbers: 625, 676, 729, 784, 841, 900
Cube numbers: 512, 729, 1000, 1331
Palindrome numbers: 949, 959, 969, 979, 989, 999, 1001, 1111, 1221, 1331

Next, we notice that

these partial sequences are all the exact same length as the words describing them. ("Palindrome" has 10 letters, and there are 10 palindrome numbers.) Turns out they even intersect in the exact same way!
enter image description here enter image description here

Take the numbers in the first sequence (which I've highlighted green) and read off their corresponding letters, and we see that this puzzle is clearly A NATURAL PROGRESSION.

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  • $\begingroup$ Dang....I was on your step two when you posted this answer... $\endgroup$ – NL628 May 12 '18 at 4:54
  • $\begingroup$ Reading the answer I turned from "Oh, well, yet another - possibly ambiguous - number sequence" to "Wow, that is really amazing and well constructed!". Great puzzle, and great answer! $\endgroup$ – Christoph May 12 '18 at 11:01

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