A Kruskal count (or a Dynkin-Kruskal count) is a sequence of entries in which each entry is based on a property of the prior term. For examples of a Kruskal count, see the post about magic
tricks that are based on this concept. Also see Wikipedia: Dynkin-Kruskal Count.
For this post, a series of Kruskal counts will be developed
using the prime omega function of the prior entry. Prime omega, sometimes
referred to as Big Omega, is the number of prime factors a number has, with
multiplicity. Prime omega of 2 is 1 since it has one prime factor. Prime omega
of 12 is 3 since it has the prime factors 2*2*3.
For this series of sequences, the first term is designated
by a(1)=m, and the formula for each subsequent term is a(n)=a(n-1)+primeomega(a(n-1)).
The base sequence of the series has m=2 and the sequence is: {2,3,4,6,8,11,12,15,17,18,21,23,24,28,31,32,37,38,40,44,47…}.
For m=3, the sequence becomes {3,4,6,8,11,12,…}. Beginning
with first term of the m=3 sequence and the second term of the base, or m=2,
sequence, the subsequent terms of the two sequences are the same. Likewise for the
m=4 sequence, the terms are the same beginning with its first term.
For m=5, the sequence becomes {5,6,8,11,12…}. In this case
the terms are same as the base sequence beginning with its second term (6).
Since for all m (conjectured - at least up to 30,000 have
been tested), sequences will match up with the base sequence, to document the
entire series of sequences, all terms of the sequences do not have to be
listed. One just needs to note at which point an m>2-sequence begins to
match up with the base sequence. For example, at m=29,052, the matching of
terms doesn’t occur until the 32nd term of the sequence. This happens to the
maximum of all the sequences up to m=30,000. To fully document the m=29,052-sequence,
one would just need to list the first 31 terms. To know subsequent terms, one could
then refer to the base sequence.
These matching points can be listed as a new sequence and
its first 86 terms are:
{1,1,1,2,1,2,1,2,2,1,1,5,4,1,3,1,1,3,2,1,2,1,1,6,2,5,1,5,4,1,1,3,3,2,2,1,1,5,1,4,3,2,1,2,2,1,1,3,2,2,5,1,1,4,2,3,1,2,1,3,2,7,1,7,6,6,5,5,1,4,3,1,1,4,1,2,3,1,1,2,9,9,8,1,8,1}.
Note that the first three terms of the base sequence is 2, 3, 4, so the
sequence above begins 1, 1, 1 because for m=2, m=3, and m=4, their first terms are
found in the base sequence. For other
starting numbers, one needs to explore higher before matching occurs. For example,
with m=13, the point at which the sequence begins to match up with the base
sequence is the 5th term.
The Mathematica program to produce the base sequence is
(producing 9,999 terms):
pseq1=NestList[#+PrimeOmega[#]&,2,10000]
This sequence is found in the On-Line Encyclopedia of
Integer Sequences (OEIS): A160649.
The Mathematica program to produce the sequence indicating
the first term at which the sequences for m=2(the base sequence itself) and
higher m’s match up with the base sequence is:
pseq={}; Do[ i=1; s=n; While[!MemberQ[pseq1, s],
s=s+PrimeOmega[s]; i++]; AppendTo[pseq, i], {n,2, 30000}];pseq
The histogram of the distribution of the terms of this sequence is:
This program produces the first 29,999 terms (note: it’s first entry starts with n=2). As noted above, the sequence begins with: {1,1,1,2,1,2,1,2,2,1,1,5,4,1,3…} and is not found in the OEIS. However, the author plans to submit it as a proposed sequence.
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