A260142 - OEIS

%I #10 Jul 19 2015 09:10:59

%S 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,

%T 24,64,864,36,216,60,160,504,192,16,288,54,6,128,24,144,1920,4,32768,

%U 32,32,216,432,8192,20,48,1296,1080,1760,4320,384,704,1728,10,360,4,2816,80

%N 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.

%C 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.

%e sigma(n)/n starts: 1/1, 3/2, 4/3, 7/4, 6/5, 2/1, 8/7, 15/8, 13/9, 9/5, ...

%e m/phi(m) starts:   1/1, 2/1, 3/2, 2/1, 5/4, 3/1, 7/6,  2/1,  3/2, 5/2, ...

%e The 1st common value is 1/1 = sigma(1)/1 = 1/eulerphi(1).

%e The 2nd common value is 3/2 = 3/eulerphi(3) = sigma(2)/2.

%e The 3rd common value is 2/1 = sigma(6)/6 = 2/eulerphi(2).

%e 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, ...

%e So this sequence begins 1, 2, 1, ...

%o (PARI) already(vsv, val, vsi, n) = {pos=vecsearch(vsv, val); if (pos, until(vsv[pos] < val, pos--); pos++; pos = vsi[pos] <= n); pos;}

%o 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););););}

%Y Cf. A259850, A260141 (numerators).

%K nonn,frac

%O 1,2

%A _Michel Marcus_, Jul 17 2015