[seqfan] Re: Comment in A003418
Eric DESBIAUX
moongerms at wanadoo.fr
Thu Sep 13 12:08:57 CEST 2012
Hello Seqfan, Mister Guninski and Mister Joerg Arndt,
Concerning my comment on A003418(n) :
For n>2, (n-1) = Sum(k=2..n, (e^((2*i*Pi)/k))^A003418(n) ).
As "Row lengths" = A003418 in A138553,
I used the same kind of formula/algorithm than for A000005
For n>0, a(n) = 1 + Sum(s=2..n, cos(Pi*n/s)^2 ).
Also tau(n) = 2 + Sum[integerpart[(cos(pi*n/k))^2], {k, 2, n-1}]. And linearisation of [(cos(pi*n/k))^2] = [1/4 * e^(-(2*i*pi*n)/k) + 1/4 * e^((2*i*pi*n)/k) + 1/2]
My goal was to have a sum of "1" for each row lengths.
Always the same pattern of 0 and 1, but from another point of view.
I'm sorry but i don't know what to do with Mister Joerg Arndt suggestion :
i write my name with the edit menu?
there is a special caracter that i don't understand in the correction?
Please help :)
Sorry for my english
and this general email on the seqfan.
Best Regards
Eric
Examples (sorry for the presentation) :
(e^((2*i*Pi)/2))^6+(e^((2*i*Pi)/3))^6 2
(e^((2*i*Pi)/2))^5+(e^((2*i*Pi)/3))^5 -1-(-1)^(1/3)
(e^((2*i*Pi)/2))^4+(e^((2*i*Pi)/3))^4 1+(-1)^(2/3)
(e^((2*i*Pi)/2))^3+(e^((2*i*Pi)/3))^3 0
(e^((2*i*Pi)/2))^2+(e^((2*i*Pi)/3))^2 1-(-1)^(1/3)
(e^((2*i*Pi)/2))^1+(e^((2*i*Pi)/3))^1 (-1)^(2/3)-1
(e^((2*i*Pi)/2))^0+(e^((2*i*Pi)/3))^0 2
(e^((2*i*Pi)/2))^0+(e^((2*i*Pi)/3))^0+(e^((2*i*Pi)/4))^0 3
(e^((2*i*Pi)/2))^1+(e^((2*i*Pi)/3))^1+(e^((2*i*Pi)/4))^1 (-1)^(2/3)+(-1+i)
(e^((2*i*Pi)/2))^2+(e^((2*i*Pi)/3))^2+(e^((2*i*Pi)/4))^2 -(-1)^(1/3)
(e^((2*i*Pi)/2))^3+(e^((2*i*Pi)/3))^3+(e^((2*i*Pi)/4))^3 -i
(e^((2*i*Pi)/2))^4+(e^((2*i*Pi)/3))^4+(e^((2*i*Pi)/4))^4 2+(-1)^(2/3)
(e^((2*i*Pi)/2))^5+(e^((2*i*Pi)/3))^5+(e^((2*i*Pi)/4))^5 -(-1)^(1/3)+(-1+i)
(e^((2*i*Pi)/2))^6+(e^((2*i*Pi)/3))^6+(e^((2*i*Pi)/4))^6 1
(e^((2*i*Pi)/2))^7+(e^((2*i*Pi)/3))^7+(e^((2*i*Pi)/4))^7 (-1)^(2/3)+(-1-i)
(e^((2*i*Pi)/2))^8+(e^((2*i*Pi)/3))^8+(e^((2*i*Pi)/4))^8 2-(-1)^(1/3)
(e^((2*i*Pi)/2))^9+(e^((2*i*Pi)/3))^9+(e^((2*i*Pi)/4))^9 i
(e^((2*i*Pi)/2))^10+(e^((2*i*Pi)/3))^10+(e^((2*i*Pi)/4))^10 (-1)^(2/3)
(e^((2*i*Pi)/2))^11+(e^((2*i*Pi)/3))^11+(e^((2*i*Pi)/4))^11 -(-1)^(1/3)+(-1-i)
(e^((2*i*Pi)/2))^12+(e^((2*i*Pi)/3))^12+(e^((2*i*Pi)/4))^12 3
(e^((2*i*Pi)/2))^0+(e^((2*i*Pi)/3))^0+(e^((2*i*Pi)/4))^0+(e^((2*i*Pi)/5))^0 4
(e^((2*i*Pi)/2))^1+(e^((2*i*Pi)/3))^1+(e^((2*i*Pi)/4))^1+(e^((2*i*Pi)/5))^1 (-1+i)+e^((2 i pi)/5)+e^((2 i pi)/3)
(e^((2*i*Pi)/2))^2+(e^((2*i*Pi)/3))^2+(e^((2*i*Pi)/4))^2+(e^((2*i*Pi)/5))^2 e^(-(2 i pi)/3)+e^((4 i pi)/5)
(e^((2*i*Pi)/2))^3+(e^((2*i*Pi)/3))^3+(e^((2*i*Pi)/4))^3+(e^((2*i*Pi)/5))^3 e^(-(4 i pi)/5)-i
(e^((2*i*Pi)/2))^4+(e^((2*i*Pi)/3))^4+(e^((2*i*Pi)/4))^4+(e^((2*i*Pi)/5))^4 2+e^(-(2 i pi)/5)+e^((2 i pi)/3)
(e^((2*i*Pi)/2))^5+(e^((2*i*Pi)/3))^5+(e^((2*i*Pi)/4))^5+(e^((2*i*Pi)/5))^5 i+e^(-(2 i pi)/3)
(e^((2*i*Pi)/2))^6+(e^((2*i*Pi)/3))^6+(e^((2*i*Pi)/4))^6+(e^((2*i*Pi)/5))^6 1+e^((2 i pi)/5)
(e^((2*i*Pi)/2))^7+(e^((2*i*Pi)/3))^7+(e^((2*i*Pi)/4))^7+(e^((2*i*Pi)/5))^7 (-1-i)+e^((2 i pi)/3)+e^((4 i pi)/5)
(e^((2*i*Pi)/2))^8+(e^((2*i*Pi)/3))^8+(e^((2*i*Pi)/4))^8+(e^((2*i*Pi)/5))^8 2+e^(-(2 i pi)/3)+e^(-(4 i pi)/5)
(e^((2*i*Pi)/2))^9+(e^((2*i*Pi)/3))^9+(e^((2*i*Pi)/4))^9+(e^((2*i*Pi)/5))^9 i+e^(-(2 i pi)/5)
(e^((2*i*Pi)/2))^10+(e^((2*i*Pi)/3))^10+(e^((2*i*Pi)/4))^10+(e^((2*i*Pi)/5))^10 1+e^((2 i pi)/3)
(e^((2*i*Pi)/2))^11+(e^((2*i*Pi)/3))^11+(e^((2*i*Pi)/4))^11+(e^((2*i*Pi)/5))^11 (-1-i)+e^((2 i pi)/5)+e^(-(2 i pi)/3)
(e^((2*i*Pi)/2))^12+(e^((2*i*Pi)/3))^12+(e^((2*i*Pi)/4))^12+(e^((2*i*Pi)/5))^12 3+e^((4 i pi)/5)
(e^((2*i*Pi)/2))^13+(e^((2*i*Pi)/3))^13+(e^((2*i*Pi)/4))^13+(e^((2*i*Pi)/5))^13 (-1+i)+e^((2 i pi)/3)+e^(-(4 i pi)/5)
(e^((2*i*Pi)/2))^14+(e^((2*i*Pi)/3))^14+(e^((2*i*Pi)/4))^14+(e^((2*i*Pi)/5))^14 e^(-(2 i pi)/5)+e^(-(2 i pi)/3)
(e^((2*i*Pi)/2))^15+(e^((2*i*Pi)/3))^15+(e^((2*i*Pi)/4))^15+(e^((2*i*Pi)/5))^15 1-i
(e^((2*i*Pi)/2))^16+(e^((2*i*Pi)/3))^16+(e^((2*i*Pi)/4))^16+(e^((2*i*Pi)/5))^16 2+e^((2 i pi)/5)+e^((2 i pi)/3)
(e^((2*i*Pi)/2))^17+(e^((2*i*Pi)/3))^17+(e^((2*i*Pi)/4))^17+(e^((2*i*Pi)/5))^17 (-1+i)+e^(-(2 i pi)/3)+e^((4 i pi)/5)
(e^((2*i*Pi)/2))^18+(e^((2*i*Pi)/3))^18+(e^((2*i*Pi)/4))^18+(e^((2*i*Pi)/5))^18 1+e^(-(4 i pi)/5)
(e^((2*i*Pi)/2))^19+(e^((2*i*Pi)/3))^19+(e^((2*i*Pi)/4))^19+(e^((2*i*Pi)/5))^19 (-1-i)+e^(-(2 i pi)/5)+e^((2 i pi)/3)
(e^((2*i*Pi)/2))^20+(e^((2*i*Pi)/3))^20+(e^((2*i*Pi)/4))^20+(e^((2*i*Pi)/5))^20 3+e^(-(2 i pi)/3)
(e^((2*i*Pi)/2))^21+(e^((2*i*Pi)/3))^21+(e^((2*i*Pi)/4))^21+(e^((2*i*Pi)/5))^21 i+e^((2 i pi)/5)
(e^((2*i*Pi)/2))^22+(e^((2*i*Pi)/3))^22+(e^((2*i*Pi)/4))^22+(e^((2*i*Pi)/5))^22
e^((2 i pi)/3)+e^((4 i pi)/5)
(e^((2*i*Pi)/2))^23+(e^((2*i*Pi)/3))^23+(e^((2*i*Pi)/4))^23+(e^((2*i*Pi)/5))^23 (-1-i)+e^(-(2 i pi)/3)+e^(-(4 i pi)/5)
(e^((2*i*Pi)/2))^24+(e^((2*i*Pi)/3))^24+(e^((2*i*Pi)/4))^24+(e^((2*i*Pi)/5))^24 3-(-1)^(3/5)
(e^((2*i*Pi)/2))^25+(e^((2*i*Pi)/3))^25+(e^((2*i*Pi)/4))^25+(e^((2*i*Pi)/5))^25 i+e^((2 i pi)/3)
(e^((2*i*Pi)/2))^26+(e^((2*i*Pi)/3))^26+(e^((2*i*Pi)/4))^26+(e^((2*i*Pi)/5))^26
e^((2 i pi)/5)+e^(-(2 i pi)/3)
(e^((2*i*Pi)/2))^27+(e^((2*i*Pi)/3))^27+(e^((2*i*Pi)/4))^27+(e^((2*i*Pi)/5))^27
e^((4 i pi)/5)-i
(e^((2*i*Pi)/2))^28+(e^((2*i*Pi)/3))^28+(e^((2*i*Pi)/4))^28+(e^((2*i*Pi)/5))^28 2+e^((2 i pi)/3)+e^(-(4 i pi)/5)
(e^((2*i*Pi)/2))^29+(e^((2*i*Pi)/3))^29+(e^((2*i*Pi)/4))^29+(e^((2*i*Pi)/5))^29 (-1+i)-(-1)^(1/3)-(-1)^(3/5)
(e^((2*i*Pi)/2))^30+(e^((2*i*Pi)/3))^30+(e^((2*i*Pi)/4))^30+(e^((2*i*Pi)/5))^30
2
(e^((2*i*Pi)/2))^31+(e^((2*i*Pi)/3))^31+(e^((2*i*Pi)/4))^31+(e^((2*i*Pi)/5))^31 (-1-i)+e^((2 i pi)/5)+e^((2 i pi)/3)
(e^((2*i*Pi)/2))^37+(e^((2*i*Pi)/3))^37+(e^((2*i*Pi)/4))^37+(e^((2*i*Pi)/5))^37 (-1+i)+e^((2 i pi)/3)+e^((4 i pi)/5)
(e^((2*i*Pi)/2))^41+(e^((2*i*Pi)/3))^41+(e^((2*i*Pi)/4))^41+(e^((2*i*Pi)/5))^41 (-1+i)+e^((2 i pi)/5)+e^(-(2 i pi)/3)
.
.
.
(e^((2*i*Pi)/2))^59+(e^((2*i*Pi)/3))^59+(e^((2*i*Pi)/4))^59+(e^((2*i*Pi)/5))^59 (-1-i)-(-1)^(1/3)-(-1)^(3/5)
(e^((2*i*Pi)/2))^60+(e^((2*i*Pi)/3))^60+(e^((2*i*Pi)/4))^60+(e^((2*i*Pi)/5))^60
4
(e^((2*i*Pi)/2))^60+(e^((2*i*Pi)/3))^60+(e^((2*i*Pi)/4))^60+(e^((2*i*Pi)/5))^60+(e^((2*i*Pi)/6))^60
5
----- Message d'origine -----
De : "Georgi Guninski" <guninski at guninski.com>
Date jeu. 13/09/2012 09:23 (GMT +02:00)
À : "seqfan at list.seqfan.eu" <seqfan at list.seqfan.eu>
Objet : [seqfan] Comment in A003418
https://oeis.org/A003418
a(0) = 1; for n >= 1, a(n) = least common multiple (or lcm) of {1, 2, ..., n}
a[x]=exp(psi(x)) where psi(x)=log(lcm(1,2,...,floor(x))) is the Chebyshev function of the second kind.
Not sure if this comment is correct:
An assertion equivalent to the Riemann hypothesis is: | log(a(n)) - n | < sqrt(n) * log(n)^2 (for n>=3).
RH implies:
|psi(x)-x| < sqrt(x) log^2 x / (8 pi) if x>73.2
Is the constant $8 pi$ missing from the comment or the comment is correct?
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