Revision 9cf72569-22c4-43b3-aa5c-d723d6a16fcb - Code Golf Stack Exchange

# [Jelly], 9 [bytes]

 ÆṪÐĿÆFL€S

[Try it online!] or [verify all test cases].

### Background

The definition of sequence [A064097] implies that 

[![definition][1]][1]

By [Euler's product formula]

[![Euler's product formula][2]][2]

where **&phi;** denotes Euler's totient function and **p** varies only over prime numbers.

Combining both, we deduce the property

[![first property][3]][3]

where **&omega;** denotes the number of distinct prime factors of **n**.

Applying the formula **k** times, where **k** is large enough so that **&phi;<sup>k+1</sup>(n) = &phi;<sup>k</sup>(n) = 1**, we get

[![second property][4]][4]

From this property, we obtain the formula

[![formula][5]][5]

where the last equality holds because **&omega;(1) = 0**.

### How it works

 ÆṪÐĿÆFL€S Main link. Argument: n

 ÐĿ Repeatedly apply the link to the left until the results are no longer
 unique, and return the list of unique results.
 ÆṪ Apply Euler's totient function.
 Since φ(1) = 1, This computes φ-towers until 1 is reached.
 ÆF Break each resulting integer into [prime, exponent] pairs.
 L€ Compute the length of each list.
 This counts the number of distinct prime factors.
 S Add the results.

[bytes]: https://github.com/DennisMitchell/jelly/wiki/Code-page
[Jelly]: http://github.com/DennisMitchell/jelly
[Try it online!]: http://jelly.tryitonline.net/#code=w4bhuarDkMS_w4ZGTOKCrFM&input=&args=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
[verify all test cases]: http://jelly.tryitonline.net/#code=w4bhuarDkMS_w4ZGTOKCrFMKw4figqxH&input=&args=NSwgMzAsIDMxLCAzMiwgMTAwLCAyMDAsIDE1NzA0Nzc4MTE2NTcyMzcyMTcwMzg0MTE1MjA2MjAzNzA0NTc3NDE0MzE5MTcxMTU4NDQxMTkzNDA4NjA4MzYzNjcwNzY3OTUzNTA4MDI0MDE2NzY4OTMzNjg5Njk5MTIxMTIzMjk5Nzk1OTY1MzM2MTgyMTUyNzMwNjk2MTg1ODYwOTU2NzYyMjI4OTg4ODgwNjUwNTU2MjU4Mzk4MzkyOTA5NTQzMjE1NzMzMDI3NTMyOTgxNjMwNzc2MTM3ODUzODgxMTE2ODM4MDQyMTc2NzYzMTA5NDU3MDAwMDY0MTM2ODAxNTU4MDE3MjY3NDM0MTU5MTM2NzM3ODUwMTgxODkzOTcyMDEwNzk5NTYxNDAxNzE4MjY4OTIyMDk0NjE5NDg4NDczNDEyNTI1NDM5NTMxOTc0NDA1MjA0MjA1ODQxOTY5NDMwNTE1MDUyMjc4Nzc2MDQ1ODY2NDk0OTUyMzc3ODk2MTEwOTI0MTc1MDk4NzYwOTk5MzU3OTYxNDY1ODQ5OTExOTg3Mzc4MjkwMTg2MTY4OTI1Njg2MzgzODEzNjY5MjExMDE2OTE3NDkyNDkzMTU1OTE0NDAxNjQxNDE1OTM1OTY2Mjc1MjAyMjY1NDM5OTY3MjgzNDYxMzQ1NTMzMDA4NzkyMDM0MjU4NTUzOTY1OTQzMjk4NDU3Ng
[A064097]: https://oeis.org/A064097
[Euler's product formula]: https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler.27s_product_formula

 [1]: https://i.sstatic.net/2zOrg.png
 [2]: https://i.sstatic.net/e1Uft.png
 [3]: https://i.sstatic.net/svLG8.png
 [4]: https://i.sstatic.net/ZV1WA.png
 [5]: https://i.sstatic.net/iZ651.png