[seqfan] A102371 Missing sloppy binary numbers - an OEIS experiment

Georg.Fischer georg.fischer at t-online.de
Thu Apr 14 21:25:51 CEST 2022

Hi Seqfans,

my initial goal was to compute A102371 as the complement of A102370,
but that is not straigtforward since A102370 is not monotone
ascending. Then I realized that A102371(n) is close to 2^n,
and looked up the differences (accidentally starting at a(3))
with this OEIS search:

That search yields 2 sequences:
A103530	a(n) = 2^n - A103529(n).
A263449	Permutation of the natural numbers: [4k+1, 4k+4, 4k+3, 4k+2, 
(and a third, A194187 Difference of n-th prime ...)

So I wondered what the differences between A103530 and A263449
could be, and with further OEIS searching and some Perl program
I finally checked that for n up to 1000 the few terms
that differ at all are 8 times of Knuth's sequence:

A182105 Number of elements merged by bottom-up merge sort.		1, 1, 2, 1, 
1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 8, 1, 1, ...

I suppose that some of you are a lot deeper in that area -
maybe you can tell whether this relation is trivial, or
can be proved, or is as miraculous as the formula in A103530?

Regards - Georg

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