# UO-Sigma PN

Mon Mar 22 08:37:06 CET 2004

```    To Neil :
Sorry, I did typo mistake.
On comment line, the following is correct.
>Some interesting part-sequences exist. s(n) : a(1),a(4), a(9), a(11),
a(15)

Isn't A000002 necessary?
I think no one understand k-values of A092356 without A000002.

Yasutoshi

>    Hello, Neil.
>    I send two sequences of UO-Sigma PN.
>
>    Yasutoshi
>
> %I A000001
> %S A000001  6, 60, 1080, 6552, 36720, 47520, 87360, 222768, 288288,
>8173440, 49585536, 683289600 , 4201148160, 6442712640,
> %T A000001  25486965504, 1610457666048, 3633511924224, 4399770343643136 ,
>9926754576979968, 629720915643477504,
> %U A000001  1420775289013796352, 1720397541537980540928,
>3881558089585691633664
> %N A000001 UO-Sigma perfecct number.
> %C A000001 If n=Product p_i^r_i then we may define the unitary ordinary
>sigma
>                   function by UO-Sigma(n)=
>UnitarySigma(2^r_1)*Sigma(n/2^r_1)
>                        =(2^r_1+1)*Product(p_i^(r_i+1)-1)/(p_i-1) ,   p_i
is
>not 2.
>                   E.g.
>                   UO-Sigma(2^4*7^2)=UnitarySigma(2^4)*Sigma(7^2)=17*57=
969
>                     So,  UO-Sigma(n) = UnitarySigma(n)        if n=2^r
>                                             = Sigma(n)      if GCD(2,n)=1
>
>                   Then an UO-Sigma perfect number satisfies UO-Sigma(n) =
>k*n for some k.
>
>                    Some interesting part-sequences exist. s(n) : a(1),
>a(4), a(9), a(11), a(1-)
>                     It has a charactor that s(n-1)|s(n).
>                     2*3, 2^3*3^2*7*13, 2^5*3^2*7*13*11,
2^7*3^2*7*11*13*43,
>2^8*3^2*7*11*13*43*257,
> %e A000001  Sequence begins  2*3, 2^2*3*5, 2^3*3^3*5, 2^3*3^2*7*13,
>2^4*3^3*5*17, 2^5*3^3*5*11, 2^6*3*5*7*13, 2^4*3^2*7*13*17,
2^5*3^2*7*13*11,
>2^6*3^2*5*7*13^2*31*61, 2^7*3^2*7*11*13*43, 2^7*3^3*5*11*43,
> %K A000001 nonn
> %O A000001 0, 1
> %A A000001 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)
> %Y A000001 A000002, A091321
>
>
>
> %I A000002
> %S A000002 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
>2, 2,
> %N A000002 k-values associated with A000001.
> %C A000002 It seems to be a constant sequence, but I conjectured that any
>positive integer more than 2 must appear.
> %K A000002 nonn
> %O A000002 0, 1
> %A A000002 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)
> %Y A000002 A000001
>
>
>
>

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