Simultaneously Inconsummate, Self and Untouchable Numbers

Today I turn 27444 days old and one property of this number is that it is inconsummate, meaning that there is no number that divided by its sum of digits equals 27444. What I noticed however, was that 27111, 27222, 27333, 27444, 27666, 27777, 27888 and 27999 are all inconsummate. 

Notice that 27555 is not inconsummate because 991980, when divided by its sum of digits (36), gives 27555. Similarly 27000 is not inconsummate because 243000, when divided by its sum of digits (9) gives 27000 as does 486000 when divided by its sum of digits (18) etc. The latter result is to be expected because doubling a number produces the same dividend when it is divided by the sum of digits. Similarly, tripling a number produces the same dividend and so any number that is "consummate" has an infinity of numbers that, when divided by their sum of digits, produce the number.

The pattern is less noticeable in the range from 26000 to 26999 where only 26111, 26666 and 26888 are inconsummate. In the range from 28000 to 28999, none of the 28XXX numbers are inconsummate. In the range from 29000 to 29999, we find 29222, 29555 and 29777 to be inconsummate. So the 27000 to 27999 millenium seems to produce one of the highest counts of ABXXX numbers but whether this is the highest, I don't know. In my post titled Inconsummate Numbers from the 1st of August 2018, I provide a list of all inconsummate numbers from 62 to 65535.

Getting back to the number associated with my diurnal age (27444) we find that it is also:

• a self number, because there is no number that, added to its sum of digits, gives 27444
• an untouchable number, because it is not equal to the sum of proper divisors of any number

One might reasonably ask the question as to how many numbers are inconsummate, self and untouchable? I was able to identify all the numbers up to 40000 with this property and there are 265 of them. Here they are:

872, 2672, 3752, 3818, 3842, 3864, 4046, 4316, 4338, 4382, 4472, 4494, 4742, 4832, 4854, 4898, 5126, 5148, 5372, 6654, 7284, 7598, 8162, 9152, 9218, 9264, 9848, 10076, 10368, 10379, 10412, 10884, 10974, 11481, 11516, 11549, 12594, 12752, 13226, 13259, 13314, 13382, 13742, 13922, 14126, 14148, 14328, 14394, 14418, 14664, 14754, 14798, 14822, 14934, 14978, 15116, 15215, 15452, 15474, 15507, 15597, 16251, 16811, 17217, 17285, 17544, 17588, 17621, 17757, 18141, 18387, 18422, 18837, 20325, 20514, 20874, 20918, 21392, 22316, 22652, 23214, 23664, 23888, 24722, 24755, 25071, 25104, 25317, 25374, 25418, 25464, 25532, 25622, 25655, 25868, 25901, 26039, 26981, 27029, 27420, 27444, 27611, 28142, 28254, 28377, 28388, 28511, 28737, 28748, 28926, 29165, 29187, 29222, 29244, 29321, 29424, 29435, 29760, 29995, 30337, 30348, 30359, 30449, 30651, 30774, 30998, 31057, 31147, 31237, 31292, 31314, 31428, 31439, 31584, 31707, 31764, 31808, 31832, 31922, 31955, 32025, 32036, 32047, 32069, 32091, 32137, 32159, 32214, 32394, 32418, 32429, 32484, 32552, 32574, 32585, 32618, 32732, 32798, 32822, 32855, 32934, 33037, 33059, 33092, 33114, 33171, 33239, 33698, 33911, 34082, 34374, 34385, 34587, 34655, 34699, 34767, 34925, 34947, 35105, 35151, 35318, 35432, 35454, 35577, 35588, 35621, 35757, 35847, 35880, 35891, 35924, 35948, 35981, 36029, 36095, 36207, 36242, 36264, 36275, 36365, 36398, 36422, 36444, 36455, 36488, 36510, 36567, 36578, 36624, 36635, 36701, 36769, 36791, 36813, 36837, 36848, 36859, 36927, 36949, 36960, 36971, 36993, 37015, 37085, 37175, 37197, 37232, 37265, 37298, 37421, 37434, 37478, 37489, 37535, 37568, 37579, 37623, 37680, 37781, 37803, 37827, 37838, 37871, 37926, 37928, 38005, 38154, 38165, 38220, 38310, 38378, 38525, 38760, 38916, 38927, 39144, 39221, 39537, 39548, 39581, 39671, 39704, 39783, 39851, 39917

So 27444 turns out to be rather special and all such numbers are in a sense quite isolated because they cannot be derived by dividing a number by its sum of digits, nor can they be had by adding the sum of a number's digits to the number and finally they cannot be derived from the addition of the proper divisors of any number. See Bespoken for Sequences entry.

Of the 265 numbers above, 20 of them are prime. These are 11549, 13259, 16811, 26981, 27611, 30449, 31147, 31237, 32069, 32159, 32429, 33037, 33911, 36791, 37489, 37579, 37781, 37871, 39581 and 39671. This is about the number you'd expect by chance, even if it is a little on the low side. There are 54 semiprimes. Of numbers with three consecutive digits that are the same, there are 23888, 27444, 29222, 29995 and 36444.