[seqfan] Re: Bell numbers of an increasing unbounded positive sequence

Vladimir Shevelev shevelev at bgu.ac.il
Sun Jan 10 19:33:31 CET 2016


Dear SeqFans,

Please, ignore the phrase from the previous
message "For example,
the Bell numbers for primes, by our definition, begin
1,1,2,4,11,23,...(Cf. the submitted A266991)."

Taking into account that, by the definition,
the minimal k-digit primes, n>=2, should consider 
in base k, i.e., with digits <=k-1, I have found that
these begin 1,1,2,4,9 (n>=0}.
For example, there are two 2-digit primes in
base 2: 2 and 3; there are four non-equivalent 
3-digit primes in base 3: 11=102_3, 13=111_3,
17=122_3, 23=212_3. Note that 19=201_3~11
=102_3; further, there exist nine non-equivalent 
4-digit primes in base 4: 67,71,73,79,83,89,101,
127,157 (with help of Peter Moses).
Further Peter found
a(5)=50, a(6)=182, a(7)=874, a(8)=3923.

Best regards,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 08 January 2016 16:37
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Bell numbers of an increasing unbounded positive sequence

The sequence of Bell numbers for palindromes
exists in OEIS. It is A188164. It is interesting
to see Bell numbers for some other classic sequences
(with less than exponential growth). For example,
the Bell numbers for primes, by our definition, begin
1,1,2,4,11,23,...(Cf. the submitted A266991).

Best regards,
Vladimir

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 07 January 2016 16:49
To: seqfan at list.seqfan.eu
Subject: [seqfan] Bell numbers of an increasing unbounded positive sequence

Dear SeqFans,

I submitted sequence A266946 "Smallest
numbers of distinct digital types of the positive
integers in base 10". If the base 10 in general to
change by base k, then let us denote by m_k(n)
the number of different digital types of n-digit
numbers in base k. Then exactly the k first
values of m_k(n), n=1,...,k, coincide with
Bell numbers A110(1),...,A110(k). One can prove
this, using the comment in A110 by Bill Blewett.
Consider now an increasing unbounded posiive
sequence A={a(n)}. Then we can consider the
smallest numbers of distinct digital types of A in
base k.  Denote by (m^A)_k(n) the number of
different digital types of n-digit numbers in A in
base k. If A contains a single-digit number, then
the union of {1,1} and  {(m^A)_k (k)}
for k=2,3,4,... is naturally called the Bell numbers
of sequence A. The first 1 formally corresponds
to n=0 in A110.  If A not contains any single-digit
number, then the union of {1,0} and  {(m^A)_k (k)}
for k=2,3,4,... is Bell sequence for A.
 For example, let A be positive palindrome in
increasing order. Then, by this definition, the
corresponding Bell numbers are
1,1,1,2,2,5,5,15,15,... (cf. in the submitted A264406).

Best regards,
Vladimir



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