A260142 Denominators 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, 2, 1, 3, 8, 4, 2, 6, 1, 24, 36, 10, 128, 12, 4, 32, 16, 12, 42, 16, 72, 384, 120, 144, 2, 24, 64, 864, 36, 216, 60, 160, 504, 192, 16, 288, 54, 6, 128, 24, 144, 1920, 4, 32768, 32, 32, 216, 432, 8192, 20, 48, 1296, 1080, 1760, 4320, 384, 704, 1728, 10, 360, 4, 2816, 80

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

Table of n, a(n) for n=1..63.

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, 2, 1, ...

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(denominator(rn), ", "); vrat = concat(vrat, rn); vsrat = vecsort(vrat, , 8), rn = ve[n]; if (!vecsearch(vsrat, rn) && (already(vssv, rn, vssi, n)), print1(denominator(rn), ", "); vrat = concat(vrat, rn); vsrat = vecsort(vrat, , 8); ); ); ); }

Crossrefs

Cf. A259850, A260141 (numerators).

Sequence in context: A320875 A265891 A248354 * A194505 A137307 A256420

Adjacent sequences: A260139 A260140 A260141 * A260143 A260144 A260145

Keyword

nonn,frac

Author

Michel Marcus, Jul 17 2015

Status

approved