The End of a Millennium

My diurnal age today is 26999 and thus the last of a millennium that began almost three years ago when I turned 26000 days old. The digits of the number have the rather special property that they add to 1:$$\frac{1}{2}+\frac{1}{6}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}=1$$This property qualifies it for membership in OEIS A091783:

A091783

Numbers with digits in non-decreasing order such that sum of the reciprocals of the digits is 1.

Such numbers are, not surprisingly, few and far between and all exclude zero. The initial members are:

1, 22, 236, 244, 333, 2488, 2666, 3366, 3446, 4444, 26999, 28888, 33999, 34688, 36666, 44488, 44666, 55555, 366999, 368888, 446999, 448888, 466688, 666666, 3999999, 4688999, 4888888, 6666999, 6668888, 7777777, 66999999, 68888999, 88888888, 999999999 

Here is a permalink to a SageMath algorithm that will generate these numbers up to 40000. If we relax the condition that the digits must be in non-decreasing order but still exclude zero, then up to 40000, the qualifying numbers are:

1, 22, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 2488, 2666, 2848, 2884, 3366, 3446, 3464, 3636, 3644, 3663, 4288, 4346, 4364, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6266, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284, 8428, 8482, 8824, 8842, 26999, 28888, 29699, 29969, 29996, 33999, 34688, 34868, 34886, 36488, 36666, 36848, 36884, 38468, 38486, 38648, 38684, 38846, 38864, 39399, 39939, 39993

These numbers constitute OEIS  A037268:

A037268

Sum of reciprocals of digits = 1.

If we relax the condition that none of the digits of the number can be zero and simply ignore the zeros when finding the reciprocal sum, then the resultant numbers constitute OEIS  A214959:

A214959

Numbers for which the sum of reciprocals of nonzero digits = 1.

The initial members of the sequence are:

1, 10, 22, 100, 202, 220, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 1000, 2002, 2020, 2036, 2044, 2063, 2200, 2306, 2360, 2404, 2440, 2488, 2603, 2630, 2666, 2848, 2884, 3026, 3033, 3062, 3206, 3260, 3303, 3330, 3366, 3446, 3464, 3602, 3620, 3636, 3644, 3663, 4024, 4042, 4204, 4240, 4288, 4346, 4364, 4402, 4420, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6023, 6032, 6203, 6230, 6266, 6302, 6320, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284, 8428, 8482, 8824, 8842, 10000, 20002, 20020, 20036, 20044, 20063, 20200, 20306, 20360, 20404, 20440, 20488, 20603, 20630, 20666, 20848, 20884, 22000, 23006, 23060, 23600, 24004, 24040, 24088, 24400, 24808, 24880, 26003, 26030, 26066, 26300, 26606, 26660, 26999, 28048, 28084, 28408, 28480, 28804, 28840, 28888, 29699, 29969, 29996, 30026, 30033, 30062, 30206, 30260, 30303, 30330, 30366, 30446, 30464, 30602, 30620, 30636, 30644, 30663, 32006, 32060, 32600, 33003, 33030, 33066, 33300, 33606, 33660, 33999, 34046, 34064, 34406, 34460, 34604, 34640, 34688, 34868, 34886, 36002, 36020, 36036, 36044, 36063, 36200, 36306, 36360, 36404, 36440, 36488, 36603, 36630, 36666, 36848, 36884, 38468, 38486, 38648, 38684, 38846, 38864, 39399, 39939, 39993

We can generalise even more by relaxing the requirement that the sum of the reciprocals must be unity and instead require it to be simply be any integer. These numbers, with zero excluded, constitute OEIS A034708: 

A034708

Numbers for which the sum of reciprocals of digits is an integer.

Clearly, the more 1's that there are in the number, the larger the integer e.g. 11111 has a sum of reciprocals of digits equal to 5. The members of this sequence, of which there are 321 members up to 40000, are:

1, 11, 22, 111, 122, 212, 221, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 1111, 1122, 1212, 1221, 1236, 1244, 1263, 1326, 1333, 1362, 1424, 1442, 1623, 1632, 2112, 2121, 2136, 2144, 2163, 2211, 2222, 2316, 2361, 2414, 2441, 2488, 2613, 2631, 2666, 2848, 2884, 3126, 3133, 3162, 3216, 3261, 3313, 3331, 3366, 3446, 3464, 3612, 3621, 3636, 3644, 3663, 4124, 4142, 4214, 4241, 4288, 4346, 4364, 4412, 4421, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6123, 6132, 6213, 6231, 6266, 6312, 6321, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284, 8428, 8482, 8824, 8842, 11111, 11122, 11212, 11221, 11236, 11244, 11263, 11326, 11333, 11362, 11424, 11442, 11623, 11632, 12112, 12121, 12136, 12144, 12163, 12211, 12222, 12316, 12361, 12414, 12441, 12488, 12613, 12631, 12666, 12848, 12884, 13126, 13133, 13162, 13216, 13261, 13313, 13331, 13366, 13446, 13464, 13612, 13621, 13636, 13644, 13663, 14124, 14142, 14214, 14241, 14288, 14346, 14364, 14412, 14421, 14436, 14444, 14463, 14634, 14643, 14828, 14882, 16123, 16132, 16213, 16231, 16266, 16312, 16321, 16336, 16344, 16363, 16434, 16443, 16626, 16633, 16662, 18248, 18284, 18428, 18482, 18824, 18842, 21112, 21121, 21136, 21144, 21163, 21211, 21222, 21316, 21361, 21414, 21441, 21488, 21613, 21631, 21666, 21848, 21884, 22111, 22122, 22212, 22221, 22236, 22244, 22263, 22326, 22333, 22362, 22424, 22442, 22623, 22632, 23116, 23161, 23226, 23233, 23262, 23323, 23332, 23611, 23622, 24114, 24141, 24188, 24224, 24242, 24411, 24422, 24818, 24881, 26113, 26131, 26166, 26223, 26232, 26311, 26322, 26616, 26661, 26999, 28148, 28184, 28418, 28481, 28814, 28841, 28888, 29699, 29969, 29996, 31126, 31133, 31162, 31216, 31261, 31313, 31331, 31366, 31446, 31464, 31612, 31621, 31636, 31644, 31663, 32116, 32161, 32226, 32233, 32262, 32323, 32332, 32611, 32622, 33113, 33131, 33166, 33223, 33232, 33311, 33322, 33616, 33661, 33999, 34146, 34164, 34416, 34461, 34614, 34641, 34688, 34868, 34886, 36112, 36121, 36136, 36144, 36163, 36211, 36222, 36316, 36361, 36414, 36441, 36488, 36613, 36631, 36666, 36848, 36884, 38468, 38486, 38648, 38684, 38846, 38864, 39399, 39939, 39993

For example, \(36613 \rightarrow \dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{6}+\dfrac{1}{1}+\dfrac{1}{3}=2\)

While the sum of reciprocals equaling unity is 26999's most distinctive property and qualifies it for membership in several OEIS sequences, it also enjoys the following interesting properties:

26999 is a Cunningham number since it can be written as \(30^3-1\). 

A number \(n\) is a Cunningham number if it can be written as \(C^{+}(b,k)=b^k+1\)  or  \(C^{-}(b,k)=b^k-1\)  for  \(b,k > 1\). The previous such number was \(26897=164^2+1\) and the next will be \(27001=30^3+1\).

26999 is a de Polignac number because none of the positive numbers \(26999 -2^k\) is a prime. These numbers (all composite) are:

1 --> 26997

2 --> 26995

3 --> 26991

4 --> 26983

5 --> 26967

6 --> 26935

7 --> 26871

8 --> 26743

9 --> 26487

10 --> 25975

11 --> 24951

12 --> 22903

13 --> 18807

14 --> 10615

26999 is a modest number because \(999|26999=26\). See my recent post titled Modest Numbers.

26999 is a Smith number  since the sum of its digits (35) coincides with the sum of the digits of its prime factors.