# [seqfan] Re: degree of integer

Sun Jun 26 20:50:01 CEST 2011

```Thanks,  Reinhard,  for a scheme which leads to an interesting criteron (in fact, we should insert 18, 30, 65, 77, 98,...).

Best

----- Original Message -----
From: Reinhard Zumkeller <reinhard.zumkeller at gmail.com>
Date: Sunday, June 26, 2011 17:07
Subject: [seqfan] Re: degree of integer
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> concerning arithmetic derivatives (A003315):
> insert 18 into
> 4,8,12,15,16,20,24,26,27,28,32,35,36,39,40,44,45,48,50,51, etc.
> and you will get http://oeis.org/A099308, which is complement of
> 4,8,12,15,16,20,24,26,27,28,32,35,36,39,40,44,45,48,50,51, etc. ==>
> http://oeis.org/A099309
> In the comment of http://oeis.org/A099307 you might find the
> desired criterion.
> Best
> Reinhard
>
>
> 2011/6/25 Vladimir Shevelev <shevelev at bgu.ac.il>
> >
> > Degree k(P) of a polynomial P(x) one can define by the number
> m(P) of iterations of derivative up to receiving 0 over the
> formula k(P)=m(P)-1+delta(P,0), where delta(P,0)=1, if P=0, and
> 0, otherwise.
> > Analogously, using A038554, one can introduce degree a(n) of
> integer n over the formula a(n)=t(n)-1+delta(n,0), where t(n) is
> the number of iterations of A038554 up to receiving 0 , delta is
> the Kronecker symbol. Since, for n>=1, A038554(n)<n, then
> t(n) always exists. So, we find  a(0)=0,a(1)=0,a(2)=1, a(3)=0,
> a(4)=2, a(5)=1, a(6)=1, a(7)=1, a(8)=3, etc.
> >    In case of derivative A003415, not all integers have a
> finite degree, and we can consider two sequences: 1) of numbers
> having A003415-finite degree ("polynomial-like numbers") and 2)
> of numbers having no A003415-finite degree. But here there is a
> problem: when a number has no A003415-finite degree? I do not
> know a criterion, although it seems that the hypothetical
> sequence of such numbers begins with
> 4,8,12,15,16,20,24,26,27,28,32,35,36,39,40,44,45,48,50,51, etc.
> >   On the other hand, since  the degrees of  the consecutive
> derivatives of polynomials monotonocally decrease, then  it is
> natural to define a polynomial-like number as a number which has
> monotonocally decreasing A003415-iterations. The sequence of
> such numbers begins with
> 1,2,3,5,6,7,9,10,11,13,14,17,19,21,22,23,25,29,31,33,34,37,38,41,42,43,46,47,49,53,...>
> > Regards,
> >
> >
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> >
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