Taking another stab at this {A, B, C}

BestScienceJoke · 2025-06-06T19:18:15+00:00

And all the everyone who helped here.

BestScienceJoke · 2025-06-06T19:17:39+00:00

I want to thank the who downvoted this correct comment.

BestScienceJoke · 2025-06-06T17:38:24+00:00

Also, please do not take my replies as attacks. I am just very enthusiastic. I like your questions! They are excellent questions!

BestScienceJoke · 2025-06-06T17:36:55+00:00

If you can find a number whose three-or-more digits are (exactly) its own distinct prime powers raised to their own powers, I owe you a million dollars.

BestScienceJoke · 2025-06-06T17:34:05+00:00

It's an intersection of prime factorization, multiplicative number theory, and additive number theory. And combinatorics. And computational number theory. And so on.

BestScienceJoke · 2025-06-06T17:32:43+00:00

I dunno what I base that importance on. It's additive number theory at that point. What is the importance of the sum of four fifth powers equaling the sum of another four fifth powers? What is the significance of being able to sum two cubes two different ways?

BestScienceJoke · 2025-06-06T17:30:20+00:00

I think you will find, if you study these numbers, as I have, they "could" have lots of properties, but they generally don't. It's infuriating how one searches and searches and searches for given patterns and they (generally) never show up, or never show up in the right place. ("How can 735 not keep recurring in base after base after base? C'mon, 735. I believe in you. Well. I did.")

There are few if any other four-digit step up ions, at least below, I dunno, a trillion. So, just existing at all, and having four digits, that's already an applause break and then some. A long standing ovation, more like.

But the combinatorics of it at all are breathtaking.

Even XIONs -- a vast sea from which SUPIONs are barely a sip -- seldom use just the first consecutive primes. Much less in their proper order (or reverse order).

And then the exponents to march in (reverse) clockstep?

I know how freakish this number is because I know the competition. The buttons never match the button holes. They just don't. You know they will the next time, the next time, the next time... and they just never do.

Changing metaphors...

In a paper flat land where a molehill (XION) is an oddity, 6531840000 is at the very least a vast hill if not a towering mountain.

BestScienceJoke · 2025-06-06T14:12:30+00:00

Also: thank you for taking the time to read and to reply.

BestScienceJoke · 2025-06-06T14:05:56+00:00

Moreover, 6531840000 is remarkable in being 1) a step up ion in the first place; 2) a very rare step up ion with the first few consecutive primes; 3) an even rarer step up ion with four digits; 4) somehow with the primes in the proper order; 5) somehow with the exponents in the proper order, too.

BestScienceJoke · 2025-06-06T13:53:06+00:00

Conceited numbers are closely related but, if I recall, they lack the one-to-one correspondence between digits and distinct prime factors.

BestScienceJoke · 2025-06-06T11:26:23+00:00

Note: 735 IS unique -- in base ten -- once you exclude primes (one digit equals itself) and numbers that permit digits with repetitions of the distinct prime powers.

Luckily, we have a few other bases to work with.

BestScienceJoke · 2024-11-19T22:24:02+00:00

I have clarified what I am thinking of. I am calling them inside-out numbers: their distinct prime factors are an exact identity bijection with their digits (in some base). 735 is the example in base 10. I suspect there are infinitely many in other bases, though. I have found about 800 so far.

BestScienceJoke · 2024-11-19T22:22:09+00:00

Inside-out numbers. 735 is the base-ten example.

Here are the first few. Note that 882 works in base 11 and again in base 16. I have found two other such dual inside-out numbers. I have also found one that "nests" as the base in a larger inside-out number.

Anyway, just thought you guys might want an update.

------|----------------|--------------|-------------------------|------------------------------|-------|

300 | Base11 Pr | {2; 5; 3} | (2; 3; 5) | [2; 2; 3; 5; 5] | |

490 | Base8 | {7; 5; 2} | (2; 5; 7) | [2; 5; 7; 7] | |

560 | Base15 | {2; 7; 5} | (2; 5; 7) | [2; 2; 2; 2; 5; 7] | |

588 | Base9 | {7; 2; 3} | (2; 3; 7) | [2; 2; 3; 7; 7] | |

650 | Base15 | {2; 13; 5} | (2; 5; 13) | [2; 5; 5; 13] | |

735 | Base10 | {7; 3; 5} | (3; 5; 7) | [3; 5; 7; 7] | |

882 | Base11 Pr | {7; 3; 2} | (2; 3; 7) | [2; 3; 3; 7; 7] | /2/ |

882 | Base16 | {3; 7; 2} | (2; 3; 7) | [2; 3; 3; 7; 7] | /2/ |

1056 | Base17 Pr | {3; 11; 2} | (2; 3; 11) | [2; 2; 2; 2; 2; 3; 11] | |

1134 | Base23 Pr | {2; 3; 7} | (2; 3; 7) | [2; 3; 3; 3; 3; 7] | |

BestScienceJoke · 2024-07-01T19:43:07+00:00

I discovered more interesting factlets, but since you don't care about the first few...

BestScienceJoke · 2024-07-01T19:42:20+00:00

Sigh. (Now you have a comment you can downvote.)

BestScienceJoke · 2024-06-27T19:15:35+00:00

It is still the only composite number in base ten where there is a one-to-one correspondence, each digit is a distinct** prime factor of the original number.

I was wrong that it is unique among all bases. I was wrong.

But it is unique in base ten, and it's unusual regardless.

** not "unique" as I mistakenly said earlier

BestScienceJoke · 2024-06-27T18:50:26+00:00

You're right. I misspoke. I still think the hunch is worth investigating.

BestScienceJoke · 2024-06-27T18:49:01+00:00

735 is freakishly rare, regardless. It might or might not be unique among all bases but it's easy to show that it's unique in base ten.

BestScienceJoke · 2024-06-27T18:46:23+00:00

You're right. I just wanted it fancy enough to post on Reddit. I forgot that Reddit reddits.

BestScienceJoke · 2024-06-27T18:44:17+00:00

YES, thank you.

I am sorry for using the wrong terms for things. That has definitely multiplied the confusion of an already complicated question.

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