Happy Numbers With Height of 7.

[Harry Neel]

Good Time of Day to all:

My understanding concerning happy numbers and their 'height' is that there is a conjecture that such numbers have a maximum height of 7. Does anyone know if that conjecture still stands, or has a greater maximum height been identified?

Regards,

Harry Neel

[Amiram Eldar]

Hi,

Sequence A001273 gives the smallest happy number of height n for n = 0 to 16.

Best,

Amiram

[Daniel Mondot]

A related sequence A018785 says that "at n=13 term becomes periodic with period 49", so one would expect the b-file to contain at least 13+49=62  terms, but it only contains 58 terms. The last 4 terms of the period are instead listed in the line for the Formula. Perhaps someone should include these in the b-file. Also the b-file should probably include 2 periods to show periodicity.

Also, a proof of the periodicity would be welcome (but considering the size of these numbers, that proof might not be simple)

Note that the fact that A018785 is periodic implies that there is no maximum height for happy numbers.

Daniel.

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