Fibonacci From Prime Factors

Gemini's Infographic Summary of the Content in this Post

I noticed that the number (\( \textbf{28075} \)) associated with my diurnal age today has an interesting property relating to its prime factors:$$28075=5^2 \times 1123$$The number has distinct prime factors of \( \textbf{5}\) and \( \textbf{1123}\). If these two factors are written in reversed order and then concatenated, the number \( \textbf{11235}\) is formed with digits that form a Fibonacci sequence:$$1+1 \rightarrow 2 \text{ and } 2 + 3 \rightarrow 5$$This got me thinking about what other numbers have this property and so I set Gemini to work to find all such numbers in the range from 1 to 40000. It turns out that the following numbers qualify (permalink):

22, 26, 30, 44, 52, 60, 66, 70, 88, 90, 101, 104, 115, 120, 132, 140, 141, 150, 158, 167, 176, 180, 198, 203, 205, 208, 210, 240, 242, 253, 257, 264, 270, 280, 300, 301, 316, 330, 338, 347, 350, 352, 360, 396, 416, 420, 423, 427, 450, 480, 484, 490, 528, 540, 560, 575, 594, 600, 611, 617, 630, 632, 660, 676, 700, 704, 720, 726, 750, 771, 790, 792, 810, 832, 835, 840, 900, 960, 968, 980, 990, 1025, 1050, 1056, 1080, 1120, 1123, 1188, 1200, 1222, 1260, 1264, 1265, 1269, 1320, 1350, 1352, 1400, 1408, 1421, 1440, 1452, 1459, 1470, 1500, 1580, 1584, 1620, 1650, 1664, 1680, 1750, 1782, 1800, 1890, 1920, 1936, 1960, 1980, 2100, 2107, 2112, 2160, 2178, 2240, 2250, 2313, 2376, 2400, 2430, 2444, 2450, 2520, 2528, 2640, 2645, 2662, 2700, 2704, 2783, 2800, 2816, 2875, 2880, 2904, 2940, 2970, 2989, 3000, 3150, 3160, 3168, 3240, 3257, 3300, 3328, 3360, 3430, 3500, 3564, 3600, 3630, 3750, 3780, 3807, 3840, 3872, 3920, 3950, 3960, 4050, 4175, 4200, 4224, 4320, 4356, 4377, 4394, 4410, 4480, 4500, 4752, 4800, 4860, 4888, 4900, 4950, 5040, 5056, 5125, 5167, 5250, 5279, 5280, 5324, 5346, 5400, 5408, 5600, 5615, 5632, 5670, 5760, 5808, 5819, 5880, 5887, 5940, 6000, 6300, 6319, 6320, 6325, 6336, 6480, 6534, 6600, 6627, 6656, 6720, 6750, 6860, 6939, 7000, 7128, 7200, 7260, 7290, 7350, 7500, 7560, 7680, 7744, 7840, 7900, 7920, 7943, 7986, 8100, 8250, 8400, 8405, 8448, 8640, 8712, 8750, 8788, 8820, 8910, 8960, 9000, 9450, 9504, 9600, 9720, 9776, 9800, 9900, 9947, 10080, 10112, 10201, 10290, 10500, 10560, 10648, 10692, 10800, 10816, 10890, 11200, 11250, 11264, 11340, 11421, 11520, 11616, 11760, 11880, 12000, 12150, 12250, 12482, 12600, 12640, 12672, 12943, 12960, 13068, 13131, 13200, 13225, 13230, 13312, 13440, 13500, 13720, 13915, 14000, 14256, 14375, 14400, 14520, 14580, 14700, 14749, 14850, 15000, 15120, 15360, 15488, 15680, 15750, 15800, 15840, 15886, 15972, 16038, 16200, 16500, 16800, 16896, 17010, 17150, 17280, 17424, 17500, 17576, 17640, 17820, 17920, 18000, 18150, 18750, 18900, 19008, 19200, 19440, 19552, 19600, 19602, 19750, 19800, 19881, 20160, 20224, 20250, 20580, 20817, 20875, 20923, 21000, 21120, 21296, 21347, 21384, 21600, 21632, 21780, 21870, 22050, 22400, 22500, 22528, 22680, 23040, 23232, 23520, 23760, 23958, 24000, 24010, 24300, 24500, 24750, 24964, 25200, 25280, 25344, 25625, 25920, 26047, 26136, 26250, 26400, 26460, 26624, 26730, 26880, 27000, 27440, 27889, 28000, 28075, 28350, 28512, 28717, 28800, 29040, 29095, 29160, 29282, 29400, 29700, 30000, 30240, 30613, 30720, 30870, 30976, 31360, 31500, 31600, 31625, 31680, 31772, 31944, 32076, 32400, 32670, 33000, 33600, 33750, 33792, 34020, 34263, 34300, 34560, 34848, 35000, 35152, 35280, 35640, 35840, 36000, 36300, 36450, 36750, 37500, 37800, 38016, 38400, 38880, 39104, 39200, 39204, 39393, 39500, 39600, 39690, 39930

Of course looking at these numbers it's not immediately apparent what the Fibonacci digit sequence is but the Gemini program creates a table to show this. I'll restrict the range to between 28000 and 29000. The result is shown in Figure 1:

Figure 1: permalink

The fact that we are only considering \( \textbf{distinct} \) prime factors helps the program run quickly and there are no problems using it with SageMathCell. However if we allow multiplicity of factors, the number of permutations increases dramatically and SageMathCell quickly times out even if we restrict the range to between 28000 and 29000. So I think working only with distinct prime factors is the way to go.