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... but there doesn't seem to be anything missing.

$0,1,2,3,4,5,6,7,1527465$

What then could cause that rather large gap?

zennehoy
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  • Is this a puzzle of your own creation? – user46002 Jan 08 '19 at 15:11
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    @Hugh I came across the sequence more or less by accident, and was surprised that it wasn't on OEIS yet. I figured I'd post it here before adding it there :) – zennehoy Jan 08 '19 at 15:20
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    They're the solutions to the polynomial x^9 - 1527493x^8 + 42769342x^7 - 49185690x^6 + 2993838169x^5 - 10339423717x^4 + 20058670380x^3 - 5040x^2 + 7698423600x = 0, but for some reason I doubt that's the cause. – Excited Raichu Jan 08 '19 at 15:39

1 Answers1

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The sequence 0,1,2,3,4,5,6,7 pointed towards octal numbers.
If we take the number 1527465 (base 10) and convert it to base 8 we get the same digits in reverse order 5647251
The gap is because no other number between 7 and 1527465 shares that property.

snetch
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rhsquared
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    So, the sequence is: "numbers in base-10 that equal themselves backwards in base-8"? What could cause this gap? – user46002 Jan 08 '19 at 15:46
  • @Hugh I've no clue yet. I'm still thinking (as much as I can do any such thing) – rhsquared Jan 08 '19 at 15:48
  • @rhaquared That's alright; I have no idea either. For a start, any number containing an 8 or a 9 is out. Maybe it's worth brute force checking all the numbers up to 1527465 with a computer. – user46002 Jan 08 '19 at 15:50
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    @Hugh see my edit – rhsquared Jan 08 '19 at 15:53
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    Don't overthink it :) – zennehoy Jan 08 '19 at 15:56
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    @Hugh just brute forced it all the way through 10,000,000 and you're right, these eight numbers are the only ones that reverse into their octal counterparts. – snetch Jan 08 '19 at 18:34
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    @snetch: Note that any such numbers must be less than 10^10, since any 10-digit number in base 10 will have at least 11 digits in base 8. – Michael Seifert Jan 08 '19 at 19:43
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    While this answer is correct, I would hate to be working on the puzzle that required finding that out. – coteyr Jan 09 '19 at 03:40
  • @MichaelSeifert The 10-digit base 10 number 1073741823 is still only 10 digits when represented in base 8. – jarnbjo Jan 09 '19 at 13:00
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    @jarnbjo: You're right; I got mixed up between the exponent and the number of digits in the number. I should have said: "any such number must be less than 10^10, since any 11-digit number in base 10 will have at least 12 digits in base 8." – Michael Seifert Jan 09 '19 at 13:06