A193995

[Harry Neel]

The subject sequence, A193995, contains a comment (which states in part), " The number 1 is the only term requiring just one term to determine if it is happy. ..."

I am not absolutely sure that I understand the comment, or if the comment is correct because it would appear that the number 1 is, by default, a happy number. (If you will, a trivial matter because 1 is already 1, and 1 has been identified, or determined, to be a 'happy number.')  

Am I mistaken or could (should) A193995 be slightly modified to be more accurate?

Regards,

H. Neel

[DONG HAOXUAN]

Dear Mr. Noe,

I hope you are well. I was recently looking at your OEIS sequence A193995 after a question was raised about the wording of its definition.

After checking the data and Mathematica code, I think the sequence itself is internally consistent, but the current wording may mix together the number of orbit terms, the number of iterations, and the number of steps actually needed to determine whether a number is happy.

I also found a couple of simple relations that may be useful additions. In particular,

A193995(n) = A099645(n) + A031176(n), for n >= 1,

and A193996 consists precisely of the eight elements of the nontrivial cycle together with the happy numbers of height 7.

I was thinking of submitting a small clarification/update to the OEIS entries, but since A193995 and A193996 are your sequences, I wanted to contact you first. Would you be happy for me to submit the proposed revisions?

Best regards,
Jason Dong

[Sean A. Irvine]

Jason,

Tony Noe is not a member of this group and is no longer active in the OEIS. I suggest you go ahead and propose your edits on the sequences concerned and the editors will consider them.

Sean.

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[DONG HAOXUAN]

Thanks, Sean. I’ll go ahead and propose the edits.
Best,
Jason

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[Sean A. Irvine]

I actually had to implement A193995 to make sure I understood what was going on here.

It comes down to what is meant in the name by "terms".

Initially, the counts for the non-happy numbers looked like they were too small by 1 to me; that is, you cannot actually tell that 2 is not a happy number until you reach 4 for a second time, which takes 10 steps not 9. But then I saw that the name says "terms" not "steps".

But if you count purely the number of distinct values (i.e. "terms") needed to determine happiness then the answer is indeed 9.

Likewise, as soon as you have "1" you know it is happy, and again that is one term.

So, no, I don't think any change in the name or comments is necessary, the definition is consistent.

Sean.

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[DONG HAOXUAN]

Thanks, Sean.

That makes sense. I had initially read “terms” as the successive terms of the trajectory, which was where the ambiguity came from for me. Interpreting it in the way you describe resolves that.

I’ll leave the existing wording alone. While checking the sequence, I did find a couple of exact relations that may be useful as additions, including

A193995(n) = A099645(n) + A031176(n), for n >= 1.

I may propose those separately as formula/comment additions rather than corrections.

Thanks again for checking this.

Best,
Jason

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[Harry Neel]

Sean,

Thank you for giving this item consideration.

Regards,

Harry

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