When Sequences Overlap

I've written in previous posts about what I've called the Factors Sequence (with and without multiplicity) and the Divisors Sequence:

• Number's Divisors to Sequence Algorithm
• Number's Factors to Sequence Algorithm 2
• Number's Factors to Sequence Algorithm 1

What I noticed about the number (28268) associated with my diurnal age today is that for the Factors Sequence with multiplicity and the Divisors Sequence both terminate where they began. Here is the Factors Sequence: 28268, 7067, 14134, 42402, 169608, 28268. Figure 1 shows a plot of the points with a logarithmic vertical axis.

Figure 1

Here is Divisors Sequence: 28268, 339216, 13568640, 106005, 1696080, 21201, 169608, 5427456, 390776832, 2035296, 28268. Figure 2 shows a plot of the points with a logarithmic vertical axis:

Figure 2

Closer inspection of both sequences shows that they share a common number apart from 28268. The number is 169608 and Figure 3 shows both plots on the same graph.

Figure 3: permalink

In the range up 40000, there are 1779 numbers with these two properties:

• the factors sequence with multiplicity and the divisors sequence terminate where they started
• the two sequences share a common member different from the start and end values

Here are the numbers (permalink) from 28000 to 40000:

28001, 28012, 28019, 28036, 28056, 28124, 28132, 28148, 28163, 28204, 28244, 28248, 28252, 28268, 28292, 28307, 28324, 28348, 28364, 28372, 28388, 28396, 28444, 28468, 28492, 28499, 28532, 28536, 28564, 28588, 28612, 28628, 28643, 28676, 28684, 28724, 28732, 28776, 28793, 28796, 28804, 28868, 28892, 28901, 28924, 28949, 28961, 28964, 28968, 28979, 28996, 29021, 29033, 29044, 29064, 29084, 29092, 29108, 29116, 29123, 29156, 29164, 29201, 29212, 29243, 29252, 29256, 29276, 29308, 29339, 29356, 29372, 29428, 29444, 29452, 29453, 29468, 29483, 29492, 29516, 29548, 29564, 29588, 29612, 29636, 29684, 29692, 29708, 29756, 29764, 29784, 29788, 29812, 29832, 29852, 29873, 29884, 29928, 29932, 29972, 30004, 30052, 30072, 30076, 30124, 30172, 30203, 30264, 30269, 30284, 30323, 30347, 30388, 30389, 30404, 30408, 30436, 30449, 30452, 30467, 30476, 30504, 30508, 30524, 30532, 30539, 30548, 30552, 30604, 30644, 30652, 30689, 30716, 30773, 30788, 30803, 30836, 30844, 30884, 30916, 30956, 30988, 31004, 31052, 31076, 31084, 31124, 31132, 31139, 31148, 31196, 31204, 31228, 31244, 31252, 31253, 31259, 31276, 31324, 31348, 31396, 31436, 31444, 31469, 31484, 31512, 31547, 31564, 31588, 31612, 31636, 31649, 31652, 31721, 31756, 31793, 31828, 31844, 31868, 31876, 31916, 31924, 31964, 31988, 31992, 31996, 32003, 32009, 32012, 32084, 32088, 32092, 32108, 32132, 32136, 32141, 32188, 32204, 32228, 32232, 32252, 32284, 32308, 32332, 32376, 32381, 32424, 32476, 32507, 32516, 32524, 32548, 32561, 32564, 32568, 32572, 32573, 32596, 32603, 32612, 32633, 32636, 32692, 32712, 32756, 32788, 32789, 32804, 32812, 32828, 32843, 32852, 32908, 32933, 32955, 32987, 32996, 33004, 33028, 33044, 33053, 33068, 33096, 33107, 33116, 33196, 33236, 33284, 33288, 33308, 33332, 33347, 33356, 33364, 33384, 33388, 33404, 33428, 33432, 33436, 33461, 33484, 33521, 33528, 33532, 33569, 33596, 33604, 33628, 33644, 33652, 33668, 33672, 33713, 33749, 33764, 33773, 33809, 33812, 33836, 33864, 33884, 33892, 33916, 33932, 33941, 33956, 33964, 33988, 34008, 34012, 34028, 34036, 34076, 34124, 34196, 34204, 34228, 34244, 34253, 34268, 34316, 34348, 34372, 34412, 34444, 34468, 34484, 34532, 34556, 34584, 34604, 34612, 34628, 34636, 34667, 34732, 34844, 34868, 34913, 34949, 34968, 34972, 34996, 35036, 35068, 35069, 35081, 35092, 35108, 35164, 35188, 35204, 35236, 35252, 35256, 35339, 35363, 35372, 35404, 35428, 35448, 35492, 35516, 35524, 35564, 35573, 35596, 35612, 35636, 35668, 35684, 35708, 35756, 35788, 35812, 35908, 35924, 35932, 35933, 35956, 35963, 35993, 36024, 36068, 36076, 36083, 36092, 36124, 36148, 36168, 36188, 36212, 36284, 36292, 36299, 36308, 36312, 36316, 36332, 36353, 36356, 36388, 36404, 36408, 36428, 36452, 36467, 36476, 36484, 36501, 36524, 36572, 36596, 36629, 36668, 36676, 36683, 36696, 36716, 36761, 36772, 36788, 36821, 36844, 36868, 36888, 36892, 36916, 36923, 36929, 36932, 36984, 36988, 37012, 37013, 37036, 37049, 37052, 37084, 37148, 37156, 37181, 37196, 37204, 37228, 37252, 37253, 37316, 37388, 37412, 37464, 37516, 37532, 37547, 37556, 37628, 37708, 37796, 37804, 37848, 37853, 37876, 37924, 37948, 37972, 38036, 38068, 38092, 38108, 38116, 38136, 38184, 38189, 38201, 38212, 38228, 38252, 38276, 38284, 38308, 38333, 38356, 38372, 38396, 38428, 38444, 38453, 38468, 38472, 38501, 38548, 38564, 38588, 38603, 38636, 38668, 38669, 38684, 38692, 38732, 38747, 38764, 38804, 38812, 38828, 38852, 38861, 38867, 38873, 38908, 38924, 38933, 39028, 39044, 39052, 39089, 39092, 39107, 39144, 39172, 39188, 39192, 39196, 39233, 39236, 39308, 39323, 39336, 39364, 39388, 39412, 39419, 39432, 39452, 39476, 39521, 39524, 39569, 39572, 39576, 39596, 39624, 39652, 39668, 39748, 39772, 39779, 39812, 39827, 39836, 39844, 39864, 39908, 39916, 39932, 39953, 39956, 39964, 39988, 39989