# [seqfan] Re: Uphi(m)=UPhi(n)=3/4*(m*n)^(1/2)

Richard Mathar mathar at strw.leidenuniv.nl
Tue Nov 4 21:42:27 CET 2008

```yk> From seqfan-bounces at list.seqfan.eu  Tue Nov  4 08:37:51 2008
yk> Date: Tue, 04 Nov 2008 15:20:21 +0900
yk> From: "koh" <zbi74583.boat at orange.zero.jp>
yk> To: seqfan at list.seqfan.eu
yk> Subject: [seqfan]  Uphi(m)=UPhi(n)=3/4*(m*n)^(1/2)
yk>
yk>     Hi, Seqfans
yk>     Definition of Amicable Number is the following.
yk>
yk>  　  Sigma(x) = Sigma(y) = x+y
yk>
yk>     If "Sigma" is replaced with the other divisor functions or "x+y" is replaced with high degree or rational or irrational formulas then it becomes generalized Amicable Number.
yk>
yk>     I calculated the examples of the following equtation.
yk>
yk>     UnitaryPhi(x) = UnitaryPhi(y) = 3/4*(x*y)^(1/2) , y<=x
yk>
yk>     x=y=2^2
yk>     x=y=2^3*7
yk>     x=y=2^4*5
yk>     x=y=2^5*5*31
yk>     x=y=2^6*3^2*7
yk>     x=y=2^6*3^3*7*13
yk>     x=y=2^7*3^2*7*127
yk>     x=y=2^7*3^3*7*13*127
yk>     x=y=2^8*5*17
yk>     x=y=2^9*3^2*7*73
yk>     x=y=2^9*3^3*7*13*73
yk>     x=y=2^10*3^3*5^2*11*13*31
yk>
yk>     x=5*3^2*11^2,      y=5*31^2
yk>     x=2^11*5*3^2*89^2, y=2^11*5*11^2*23^2
yk>
yk>     Could anyone do exhausitive search?
yk>     I would like to know the other examples of the case not{x=y}.
yk>
yk>
yk>
yk>
yk>     %I A000001
yk>     %S A000001 4, 56, 80, 4032, 4960, 5445, 21760, 157248, 1024128, 39940992, 2354688, 91832832, 729999360
yk>     %N A000001 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(m)=3/4*(m*n)^(1/2), n<=m

There is an obvious typo in this definition (?!). I am also strongly in favor of including
explicit OEIS A-number references to rare functions such as UnitaryPhi(.) into
submissions of that kind (see the discussion in the seqfan list of Oct 2006).

I ran the following Maple prog and confirm that this list is
complete in the range y<=x<=21760. There are no further values
<= 21760 in the sequence. A Java version is available on request.

A047994 := proc(n) option remember ;
local ifacts;
ifacts := ifactors(n)[2] ;
mul( op(1,op(i,ifacts))^op(2,op(i,ifacts))-1,i=1..nops(ifacts)) ;
end:
x := 1:
while true do
uphix := A047994(x) ;
if uphix mod 3 = 0 then
for y from 1 to x do
uphiy := A047994(y) ;
if uphix^2 = uphiy^2 and uphix^2 = 9/16*x*y then
print(x,ifactors(x)[2],y,ifactors(y)[2]) ;
fi ;
od :
fi:
x := x+1 :
od :

yk>     %C A000001 a(6) and a(13) are the cases not{n=m}
yk>     %e A000001 Factorization of cases not{n=m}
yk>                m,n=5*3^2*11^2,5*31^2
yk>                m.n=2^11*5*3^2*89^2,2^11*5*11^2*23^2
yk>     %Y A000001 A000002
yk>     %K A000001 none
yk>     %O A000001 0,1
yk>     %A A000001 Yasutsohi Kohmoto zbi74583.boat at orange.zero.jp
yk>
yk>     %I A000002
yk>     %S A000002 4, 56, 80, 4032, 4960, 4805, 21760, 157248, 1024128, 39940992, 2354688, 91832832, 655452160
yk>
yk>     %N A000002 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(m)=3/4*(m*n)^(1/2), n<=m

There is the obvious typo in the definition again. Of course it is puzzling that
the definition is the same as A00001 above, although the numbers are different (?!).

yk>     %C A000002 a(6) and a(13) are the cases not{n=m}
yk>     %Y A000002 A000001
yk>     %K A000002 none
yk>     %O A000002 0,1
yk>     %A A000002 Yasutsohi Kohmoto zbi74583.boat at orange.zero.jp
--
Richard J Mathar
The Netherlands                 URL http://www.strw.leidenuniv.nl/~mathar

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