[seqfan] Re: Moebius-transform of A005187 ?

[Antti Karttunen]

On Sat, Mar 10, 2018 at 8:33 PM, Antti Karttunen <antti.karttunen at gmail.com>
wrote:

>
>
> On Sat, Mar 10, 2018 at 8:10 PM, Antti Karttunen <
> antti.karttunen at gmail.com> wrote:
>
>>
>> Consider the sequences A035532 and A297111:
>>
>> https://oeis.org/search?q=+%091%2C+2%2C+3%2C+4%2C+7%2C+4%2C+
>> 10%2C+8%2C+12%2C+8%2C+18%2C+8%2C+22%2C+12&sort=&language=&go=Search
>>
>> The latter is Moebius-transform of A005187 I recently submitted, while
>> from the formula-section of the former: G.f.: Sum A005187(n) x^n = Sum
>> a(n)*x^n/(1-x^n)
>> it seems to me that it also claims to be Möbius-transform of A005187.
>> (See e.g. https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula#
>> Series_relations )
>> However, the sequences differ for the first time at n=15, although even
>> after that they seem to contain many identical terms.
>>
>> (Moreover, is there an indexing mismatch in Zumkeller's formula a(n) =
>> 2*A000010(n) - A010051(n)*A048881(n) ?)
>>
>>
>> Now, I just wonder where this "slightly flawed" insight in A035532 stems
>> from, and on what specific n it fails: 15, 21, 25, 27, 33, 35, 51,  ...
>> (where two sequences differ, "manually tallied" so beware) and why?
>>
>
Actually, these should be the nonzero values obtained by composite n in:
https://oeis.org/A297115 Möbius transform of A000120, binary weight of n.
Of course for primes p, A297115(p) = A000120(p)-1. So it is at composite
n=15 where the claim in A035532 no longer holds.

Best regards,

Antti
(and sorry for the monologue).




>>
> My own guess: The original author had computed only the initial 14 terms
> of the sequence, then asked Superseeker or a similar program what else it
> might be, and obtained the GF present in the formula-section.
>
> Antti
>
>
>>
>> Best regards,
>>
>> Antti
>>
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