<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-2022-jp">
<META content="MSHTML 6.00.2900.2180" name=GENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY bgColor=#ffffff>
<DIV><FONT face="MS UI Gothic"> Neil.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> I submitted sequences of
Perfect Number using the following divisor functions.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> But I think I didn't submitted
these divisor functions.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> I suppose OEIS should have
them too</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %I
A000001<BR> %S A000001 1, 3, 4, 7, 6, 12, 8, 15, 10, 18, 12,
28, 14, 24, 24, 31, 18, 30, 20, 42<BR> %N A000001
OrdinaryUnitarySigma(n) : If n=Product p_i^r_i then
OUSigma(n)=Sigma(2^r_1)*UnitarySigma(n/2^r_1)=(2^(r_1+1)-1)*Product(p_i^r_i+1),
p_i is not 2.<BR>
<BR> %e
A000001
OUSigma(2^4*7^2)=Sigma(2^4)*UnitarySigma(7^2)=31*50=1550.<BR>
%O A000001 1,2 <BR> %Y
A000001 A000002, A091321<BR> %K
A000001 nonn, mult, <BR> %A
A000001 Yasutoshi Kohmoto (<A
href="mailto:zbi74583@boat.zero.ad.jp">zbi74583@boat.zero.ad.jp</A>)</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %I
A000002<BR> %S A000002 1, 3, 4, 5, 6, 12, 8, 9, 13, 18, 12,
20, 14, 24, 24, 17, 18, 39, 20, 30 <BR> %N A000002
UnitaryOrdinarySigma(n) : If n=Product p_i^r_i then
UOSigma(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.<BR> %e
A000002
UOsigma(2^4*7^2)=UnitarySigma(2^4)*sigma(7^2)=17*57=969<BR> %O
A000002 1,2 <BR> %Y
A000002 A000001, A092356
<BR> %K A000002 nonn, mult,
<BR> %A A000002 Yasutoshi Kohmoto (<A
href="mailto:zbi74583@boat.zero.ad.jp">zbi74583@boat.zero.ad.jp</A>)<BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> Yasutoshi
</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> PS</FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> Two divisor functions in my
May,25 mail don't appear on OEIS.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> Did you see the
mail?</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %I
A000001<BR> %S A000001 1, 4, 5, 8, 7, 20, 9, 16, 14, 28, 13,
40, 15, 36, 35, 32, 19, 56, 21, 56<BR> %N A000001
(+2)Sigma(n) : If n=Product p_i^r_i then (+2)Sigma(n)=Product (2+Sum
p_i^s_i, s_i=1 to r_i)=Product(1+(p_i^(r_i+1)-1)/(p_i-1)) ,
<BR>
(+2)Sigma(1)=1 <BR> %e
A000001
(+2)Sigma(6)=(2+2)*(2+3)=20.<BR> %O
A000001 1,2 <BR> %Y
A000001 A000002, A052396<BR> %K
A000001 nonn, mult, <BR> %A
A000001 Yasutoshi Kohmoto (<A
href="mailto:zbi74583@boat.zero.ad.jp">zbi74583@boat.zero.ad.jp</A>)
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %I
A000002<BR> %S A000002 1, 4, 5, 6, 7, 20, 9, 10, 11, 28, 13,
30, 15, 36, 35, 18, 19, 44, 21, 42<BR> %N A000002
(+2)UnitarySigma(n) : If n=Product p_i^r_i then (+2)Sigma(n)=Product
(2+p_i^r_i) ,
<BR>
(+2)UnitarySigma(1)=1 <BR> %e
A000002
(+2)UnitarySigma(12)=(2+3)*(2+4)=30.<BR> %O
A000002 1,2 <BR> %Y
A000002 A000001, A054862<BR> %K
A000002 nonn, mult, <BR> %A
A000002 Yasutoshi Kohmoto (<A
href="mailto:zbi74583@boat.zero.ad.jp">zbi74583@boat.zero.ad.jp</A>)<BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> </FONT></DIV></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> <FONT size=2> </FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"><FONT
size=2> </DIV></FONT></FONT></BODY></HTML>