## Weighted roundness

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• A new concept I ve been playing with: If n=p_1^e_1...p_k^e_k, define wr(n)=sigma(i*e_1) and the quadratic version: wr(n)=sigma(i^2*e_1) (even generally)
Message 1 of 1 , Dec 31, 2002
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A new concept I've been playing with:

If n=p_1^e_1...p_k^e_k, define

wr(n)=sigma(i*e_1)

wr(n)=sigma(i^2*e_1)

(even generally)

wr(n,k)=sigma(i^k*e_1)

It is assumed that the p's are ordered, hence the weighting system.

Some code and results:

weightedroundness(n)=local(f,fl,s);f=factor(n);fl=length(f[,1]);s=0;for
(i=1,fl,s=s+i*f[,2][i]);s
alias(wr,weightedroundness)
for (j=2,500,print1(wr(j)","))
1,1,2,1,3,1,3,2,3,1,4,1,3,3,4,1,5,1,4,3,3,1,5,2,3,3,4,1,6,1,5,3,3,3,6,1,3,3,
5,1,6,1,4,4,3,1,6,2,5,3,4,1,7,3,5,3,3,1,7,1,3,4,6,3,6,1,4,3,6,1,7,1,3,5,4,3,
6,1,6,4,3,1,7,3,3,3,5,1,8,3,4,3,3,3,7,1,5,4,6,1,6,1,5,6,3,1,8,1,6,3,6,1,6,3,
4,4,3,3,8,2,

weightedroundness2(n)=local(f,fl,s);f=factor(n);fl=length(f[,1]);s=0;for
(i=1,fl,s=s+i^2*f[,2][i]);s
alias(wr2,weightedroundness2)
for (j=2,500,print1(wr2(j)","))
1,1,2,1,5,1,3,2,5,1,6,1,5,5,4,1,9,1,6,5,5,1,7,2,5,3,6,1,14,1,5,5,5,5,10,1,5,
5,7,1,14,1,6,6,5,1,8,2,9,5,6,1,13,5,7,5,5,1,15,1,5,6,6,5,14,1,6,5,14,1,11,1,
5,9,6,5,14,1,8,4,5,1,15,5,5,5,7,1,18,5,6,5,5,5,9,1,9,6,10,1,14,1,7,14,5,1,14
,1,14,5,8,1,

This is the differences between the wr_2 and wr_1:

for (j=2,500,print1(wr2(j)-wr(j)","))
0,0,0,0,2,0,0,0,2,0,2,0,2,2,0,0,4,0,2,2,2,0,2,0,2,0,2,0,8,0,0,2,2,2,4,0,2,2,
2,0,8,0,2,2,2,0,2,0,4,2,2,0,6,2,2,2,2,0,8,0,2,2,0,2,8,0,2,2,8,0,4,0,2,4,2,2,
8,0,2,0,2,0,8,2,2,2,2,0,10,2,2,2,2,2,2,0,4,2,4,0,8,0,2,8,2,0,6,0,8,2,2,0,8,2
,2,2,2,2,8,0

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths/