[seqfan] A144648
zbi74583.boat at orange.zero.jp
zbi74583.boat at orange.zero.jp
Mon Feb 2 05:48:11 CET 2009
Hi, Seqfans
I explain the reason why these terms of A144648 and A144651 are complete and
all.
To search all of the ?Seed? of Amicable number is much easier than to search
all of Amicable number.
Seed is defined as follows.
If numbers m,n satisfy the following equation then m,n is called Seed.
Sigma(m) = Sigma(n)
Amicable number {m,n} is also Seed.
But the most useful Seed must satisfy the following condition.
GCD(m,n) = 1
Example : {2^2*5*11, 2^2*71} is an Amicable number. And {5*11,71} is a Seed.
If an Amicable number is represented as {c*x, c*y} where c=GCD{m,n} then
number c is called ?Sprout?.
The ideas ?Seed? and ? Sprout? are generalized for the other kind of
?Amicable number? which have the other divisor functions for example
UnitarySigma(m), UnitaryPhi(m), InfinitarySigma(m), (-1)Sigma(m),
USUP(m), etc, in the definition.
In the case of A144648 the Seed satisfies the following.
Unitary(m) = Unitary(n)
{2^2*3^2,5^2} and {2^4*3^2,11^2} are Seeds for A144648 but they have no
Sprout.
{3^2*11^2,31^2} is the smallest Seed. So the first term is the smallest
example.
Unitary(m) = Unitary(n) = (3*m^(1/2) ? 2*n^(1/2))^2 is difficult condition
and I have searched almost all Seed up to 10^4 and no other example
existed.
So I think that they are complete and all.
Yasutoshi
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