[seqfan] Re: phi-partitions of n

[Neil Sloane]

If you omit the primes, you get a more manageable sequence,
A283320, which I created the other day.

The phi-partition entries, by the way, are A283528 and A283530.
Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Wed, Mar 15, 2017 at 11:37 AM, M. F. Hasler <seqfan at hasler.fr> wrote:
> On Thu, Mar 9, 2017 at 1:55 PM, Neil Sloane wrote:
>
>> Dear Seq Fans, Call a partition n = a1+...+ak  a phi-partition
>> if phi(n) = phi(a1)+...+phi(ak), where phi() = A10().
>> There are several refinements, such as reduced phi-partitions.
>> Are these sequences (the number of ... of n) in the OEIS?
>> Also the related sequence of semisimple integers (Wang and Wang)
>
> - it seems the smallest is 746130, so this is definitely not yet in the
>> OEIS - how does it continue?
>>
>
> Actually, the cited paper,
> http://www.fq.math.ca/Papers1/44-2/quartwang02_2006.pdf
> show that 746130 is *not* semisimple, by listing its reduced phi-partitions.
> Then it says that  p9*p8*A6 (= prime(9)# / prime(7) = 13123110)
> is the smallest semisimple integer
> * *of the given form, **with a = 1 and k >= 2 **.
>
> The "given form" is   n = a  q_1 ... q_k A_i ,  where:
>   A_i = product( prime(j), j <= i ) = A002110 <https://oeis.org/A002110>(i),
> i >= 1,
>   the q's are  k >= 0  distinct primes > P := prime( i+1 ), and
>   a is a positive integer such that  a (q_1 - P) ... (q_k - P) < P
>
> If we take k=0 there is no constraint except a < P.
> So, in addition to 9 and all primes which are also semisimple, all
> multiples of the primorials A_i less than P*A_i = A_{i+1} are also
> semisimple.
> These are
> 1,2,4,6,12,18,24,30,60,90,120,150,180,210,420,630,840,... =
> A060735 <https://oeis.org/A060735> :
> Where n / (phi(n) + 1) increases.
> (... this sequence is a primorial (A002110 <https://oeis.org/A002110>)
> followed by its multiples until the next primorial, then the multiples of
> that primorial and so on.)
>
> If we add the primes and 9, we get
> 12,3,4,5,6,7,9,11,12,13,17,18,19,23,24,29,30,31,37,41,43,47,53,59,60,61,67,
> 71,73,79,83,89,90,97,101,103,107,109,113,120,127,131,137,139,149,150,...
> which seems not in the OEIS.
>
> We could call "nontrivial" the semisimple integers not of this form,
> i.e., which can only be written in the given form with k >= 1.
>
> In the case k=1, we have multiples n = a q A_i such that  a*(q - P) < P :=
> prime(i+1).
> As the paper states, a = 1, q = prime(i+2) always yields a solution
> (since prime(i+2) < 2 prime(i+1) for all i), so these could also be
> considered as "trivial" solutions. They include n = 5*2 = 10, 7*3*2 = 42,
> 11*5# = 330, ...
>
> The first "strictly nontrivial" solutions are:
> For i = 2, P = 5 > a*(q-5) with q = 7, a = 2: n = 2*7*3*2 = 84.
> For i = 3, P = 7 > a*(q-7) with q = 13, a = 1: n = 1*13*5# = 390.
> For i = 4, P = 11 > a*(q - 11) with
>  q = 13, a = 2,3,4,5 ; n = a*13*7# = a*2730
>  q = 17 and 19, a = 1 : n = 17*7# = 3570  and n = 19*7# = 3990.
>
> There are many more solutions before the first solutions with k=2.
> Concerning these, one can easily check that
>  (prime(i+2)-prime(i+1))*(prime(i+3)-prime(i+1)) < prime(i+1)
> only for  i >= 6 (but not for i=7,8,10,14,22,23,29,45),
> so  prime(8)*prime(9)*prime(6)# = prime(9)# / prime(7) is indeed the
> smallest *of this form*.
> I have submitted the set of all semisimple integers as
> https://oeis.org/draft/A283736/.
> Since they are quite dense (8, 14, 15, 16 are the only missing numbers
> below 20),
> I could also submit the subset of nontrivial (or "strictly nontrivial")
> solutions, if other editors agree.
>
> PS: I hope all SeqFans had a nice pi-day!
>
> -Maximilian
>
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