|
| |
|
|
A260141
|
|
Numerators of the distinct common values of sigma(n)/n and m/phi(m) in the order which they occur when n and m increase.
|
|
1
|
|
|
|
1, 3, 2, 7, 15, 7, 5, 13, 3, 65, 91, 31, 255, 31, 13, 85, 31, 35, 127, 51, 217, 1105, 403, 403, 7, 73, 221, 2555, 127, 635, 217, 527, 1651, 595, 33, 949, 133, 19, 267, 77, 511, 6851, 11, 65535, 89, 119, 665, 1397, 21845, 77, 143, 4123, 3937, 6141, 15841, 1157, 2047, 5621, 33, 1397, 15, 6141, 267
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
To be considered as common, a value must have appeared for some N in both sequences sigma(n)/n (A017665/A017666) and n/eulerphi(n) (A109395/A076512), with 1<=n<=N.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
sigma(n)/n starts: 1/1, 3/2, 4/3, 7/4, 6/5, 2/1, 8/7, 15/8, 13/9, 9/5, ...
m/phi(m) starts: 1/1, 2/1, 3/2, 2/1, 5/4, 3/1, 7/6, 2/1, 3/2, 5/2, ...
The 1st common value is 1/1 = sigma(1)/1 = 1/eulerphi(1).
The 2nd common value is 3/2 = 3/eulerphi(3) = sigma(2)/2.
The 3rd common value is 2/1 = sigma(6)/6 = 2/eulerphi(2).
The sequence of ratios begin: 1, 3/2, 2, 7/3, 15/8, 7/4, 5/2, 13/6, 3, 65/24, 91/36, 31/10, 255/128, 31/12, ...
So this sequence begins 1, 3, 2, ...
|
|
|
PROG
|
(PARI) already(vsv, val, vsi, n) = {pos=vecsearch(vsv, val); if (pos, until(vsv[pos] < val, pos--); pos++; pos = vsi[pos] <= n); pos; }
lista(nn) = {vrat = [1]; vsrat = [1]; ve = vector(nn, k, k/eulerphi(k)); vs = vector(nn, k, sigma(k)/k); vesv = vecsort(ve); vesi = vecsort(ve, , 1); vssv = vecsort(vs); vssi = vecsort(vs, , 1); print1(1, ", "); for (n=2, nn, rn = vs[n]; if (!vecsearch(vsrat, rn) && (already(vesv, rn, vesi, n)), print1(numerator(rn), ", "); vrat = concat(vrat, rn); vsrat = vecsort(vrat, , 8), rn = ve[n]; if (!vecsearch(vsrat, rn) && (already(vssv, rn, vssi, n)), print1(numerator(rn), ", "); vrat = concat(vrat, rn); vsrat = vecsort(vrat, , 8); ); ); ); }
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,frac
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
