It was back when I turned 26671 days old on April 11th 2022 that I first came across an unusual function that when applied repeatedly, so that the output becomes the new input, leads to zero or a loop. Here is the function \( n \) is any integer \( \ge 1\):$$\lceil \sqrt{n} \, \rceil \times (\lceil \sqrt{n}\, \rceil^2 – n) $$It can be seen that, with when \( n \) is a square number, the value of the expression is zero. When applied to most numbers, the iteration leads to zero but, far less frequently, the sequence of numbers generated by the iteration leads to a loop. 26671 is one such number. It has the following trajectory:
26671, 36900, 67357, 63180, 81648, 42328, 22248, 37800, 43875, 47250, 59732, 71785, 10452, 16171, 27264, 48472, 81549, 70642, 30324, 52675, 51750, 53352, 2079, 1702, 2604, 5200, 9417, 18326, 23120, 44217, 64144, 94488, 115808, 161293, 125022, 104076, 81719, 22022, 26671
Thus we end up where we began, but this is not always the case as we shall see. The reason that I was reminded of this function is that today I turned 26710 days old and this number also has the property that it does not end in zero under repeated iterations but instead enters a loop. In the case of 26710, the loop is:
26710, 30504, 21175, 20586, 21600, 1323, 1702, 2604, 5200, 9417, 18326, 23120, 44217, 64144, 94488, 115808, 161293, 125022, 104076, 81719, 22022, 26671, 36900, 67357, 63180, 81648, 42328, 22248, 37800, 43875, 47250, 59732, 71785, 10452, 16171, 27264, 48472, 81549, 70642, 30324, 52675, 51750, 53352, 2079, 1702
Here it can be seen that the number does not return to its starting point but instead enters a loop beginning and ending with 1702. Interestingly, 26709 also enters a loop as well. The loop is:
26709, 30668, 54208, 18873, 23598, 18172, 7155, 5950, 10452, 16171, 27264, 48472, 81549, 70642, 30324, 52675, 51750, 53352, 2079, 1702, 2604, 5200, 9417, 18326, 23120, 44217, 64144, 94488, 115808, 161293, 125022, 104076, 81719, 22022, 26671, 36900, 67357, 63180, 81648, 42328, 22248, 37800, 43875, 47250, 59732, 71785, 10452
Such pairs are not all that common. The pairs up to this point are as follows:
Overall, the numbers that do not become zero constitute about 1.94% of the numbers in the range between 1 and 26710. These numbers constitute OEIS A219960 and the members up to 26710 are:
366, 680, 691, 1026, 1136, 1298, 1323, 1417, 1464, 1583, 1604, 1702, 2079, 2125, 2222, 2223, 2374, 2507, 2604, 2627, 2821, 2844, 2897, 3152, 3157, 3159, 3183, 3210, 3231, 3459, 3697, 3715, 3762, 3802, 3866, 3888, 3936, 3948, 4004, 4111, 4133, 4145, 4231, 4299, 4388, 4414, 4614, 4653, 4683, 4685, 4780, 4794, 4815, 5004, 5025, 5084, 5103, 5130, 5193, 5200, 5244, 5342, 5382, 5453, 5509, 5513, 5515, 5524, 5529, 5558, 5707, 5793, 5832, 5877, 5888, 5902, 5950, 5980, 5989, 6015, 6103, 6129, 6205, 6295, 6310, 6335, 6447, 6469, 6489, 6498, 6513, 6522, 6662, 6676, 6767, 6788, 6956, 7009, 7025, 7063, 7095, 7152, 7155, 7200, 7217, 7258, 7261, 7397, 7408, 7410, 7420, 7422, 7452, 7460, 7463, 7469, 7575, 7625, 7751, 7937, 7942, 7947, 7971, 8020, 8043, 8112, 8150, 8163, 8237, 8250, 8335, 8383, 8399, 8400, 8407, 8503, 8621, 8700, 8762, 8785, 8794, 8848, 8947, 8971, 9141, 9175, 9222, 9234, 9332, 9352, 9417, 9452, 9483, 9499, 9663, 9754, 9763, 9780, 9841, 9913, 9916, 9928, 9948, 10031, 10118, 10126, 10134, 10179, 10211, 10221, 10232, 10245, 10269, 10290, 10357, 10431, 10452, 10472, 10546, 10673, 10738, 10766, 10835, 10844, 10851, 10866, 10902, 10927, 10945, 11050, 11077, 11083, 11086, 11149, 11166, 11238, 11246, 11404, 11419, 11457, 11458, 11460, 11464, 11551, 11595, 11610, 11628, 11729, 11794, 11858, 11868, 11921, 12025, 12204, 12411, 12465, 12469, 12574, 12606, 12661, 12716, 12775, 12784, 12789, 12821, 12894, 12915, 12931, 12939, 12950, 12951, 12963, 12987, 12997, 13019, 13173, 13327, 13381, 13465, 13475, 13512, 13578, 13602, 13643, 13662, 13670, 13722, 13770, 13833, 13913, 13966, 13980, 14007, 14073, 14111, 14189, 14220, 14330, 14340, 14459, 14466, 14543, 14662, 14670, 14673, 14731, 14801, 14872, 14881, 14896, 14964, 15024, 15097, 15130, 15195, 15217, 15335, 15355, 15379, 15406, 15559, 15564, 15608, 15668, 15731, 15891, 15900, 16171, 16191, 16218, 16338, 16388, 16417, 16438, 16505, 16525, 16549, 16551, 16568, 16586, 16681, 16695, 16707, 16715, 16815, 16843, 16854, 16860, 16975, 17070, 17164, 17170, 17461, 17474, 17539, 17544, 17577, 17648, 17718, 17728, 17763, 17878, 17882, 17972, 18008, 18026, 18065, 18123, 18139, 18172, 18187, 18270, 18326, 18334, 18367, 18402, 18419, 18423, 18491, 18534, 18546, 18666, 18716, 18854, 18873, 18882, 18945, 18958, 18965, 18990, 19005, 19006, 19127, 19253, 19285, 19330, 19356, 19540, 19547, 19674, 19677, 19686, 19690, 19716, 19735, 19847, 19848, 19853, 19894, 19950, 19972, 20156, 20187, 20195, 20206, 20209, 20295, 20345, 20421, 20524, 20554, 20583, 20586, 20686, 20709, 20749, 20803, 20892, 20899, 20965, 21121, 21175, 21223, 21248, 21324, 21332, 21426, 21451, 21522, 21539, 21600, 21618, 21622, 21627, 21721, 21837, 21857, 21929, 22009, 22020, 22022, 22032, 22035, 22114, 22153, 22164, 22248, 22254, 22295, 22356, 22367, 22394, 22442, 22444, 22445, 22452, 22577, 22813, 22903, 22945, 22995, 23006, 23118, 23120, 23138, 23205, 23221, 23226, 23265, 23287, 23303, 23319, 23333, 23470, 23573, 23597, 23598, 23639, 23648, 23690, 23789, 23836, 24050, 24116, 24168, 24269, 24284, 24352, 24366, 24392, 24441, 24546, 24704, 24711, 24734, 24793, 24817, 24874, 24895, 24908, 24946, 25038, 25072, 25076, 25089, 25090, 25129, 25157, 25175, 25176, 25179, 25181, 25194, 25223, 25236, 25320, 25336, 25465, 25555, 25640, 25675, 25698, 25708, 25727, 25742, 25743, 25834, 25862, 25930, 25945, 26106, 26108, 26159, 26187, 26198, 26208, 26220, 26306, 26456, 26479, 26506, 26509, 26519, 26526, 26650, 26665, 26671, 26709, 26710
There are a number of conjectures associated with this ceiling function. These are listed in the OEIS comments and are:
Conjecture 1: All numbers under the iteration reach 0 or, like the elements of this sequence, reach a finite loop, and none expand indefinitely to infinity.
Conjecture 2: There are an infinite number of such finite loops, though there is often significant distance between them.
Conjecture 3: There are an infinite number of pairs of consecutive integers.
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