Original entry on oeis.org
1, 3, 8, 14, 15, 21, 26, 40, 130, 144, 182, 255, 310, 372, 465, 468, 680, 980, 1524, 2170, 2210, 2418, 2448, 4030, 4536, 7008, 7956, 8890, 9906, 10220, 10416, 10668, 12648, 16335, 16660, 17082, 20216, 24624, 30800, 36792, 41106, 44055, 48400, 65535, 77112, 78320, 85120, 97790, 143000, 149688
Offset: 1
a(4) = 14 is a term because 14 = A259850(5) is the first term of A259850 whose set of prime factors is {2,7}.
28 = A259850(11) is not a term because it has the same set {2,7} of prime factors as 14.
-
R:= NULL: count:= 0: V:= {}: S:= {}:
for k from 1 while count < 50 do
V:= V union {numtheory:-sigma(k)/k};
if member(k/numtheory:-phi(k), V) then
s:= numtheory:-factorset(k);
if not member(s,S) then
R:= R, k; count:= count+1; S:= S union {s}
fi fi;
od:
R;
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.
Original entry on oeis.org
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
Offset: 1
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, ...
-
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););););}
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.
Original entry on oeis.org
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
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, ...
-
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););););}
Showing 1-3 of 3 results.
Comments