Pronic Determinants of Circulant Matrices

Consider the number 28235 that is my diurnal age today. It has a circulant matrix with a determinant 10100 that is a pronic number since 10100 = 100 x 101.$$\begin{bmatrix}

2 & 8 & 2 & 3 & 5 \\

8 & 2 & 3 & 5 & 2 \\

2 & 3 & 5 & 2 & 8 \\

3 & 5 & 2 & 8 & 2 \\

5 & 2 & 8 & 2 & 3

\end{bmatrix}$$What's interesting is that most of the permutations of the digits of 28235 have determinants of their circulant matrices that are also pronic (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22358        | 15500           | 124 x 125      
22385        | 10100           | 100 x 101      
22538        | 19100           |                
22583        | 10100           | 100 x 101      
22835        | 19100           |                
22853        | 15500           | 124 x 125      
23258        | 19100           |                
23285        | 19100           |                
23528        | 10100           | 100 x 101      
23582        | 15500           | 124 x 125      
23825        | 15500           | 124 x 125      
23852        | 10100           | 100 x 101      
25238        | 15500           | 124 x 125      
25283        | 15500           | 124 x 125      
25328        | 10100           | 100 x 101      
25382        | 19100           |                
25823        | 19100           |                
25832        | 10100           | 100 x 101      
28235        | 10100           | 100 x 101      
28253        | 10100           | 100 x 101      
28325        | 15500           | 124 x 125      
28352        | 19100           |                
28523        | 19100           |                
28532        | 15500           | 124 x 125      
32258        | 10100           | 100 x 101      
32285        | 15500           | 124 x 125      
32528        | 15500           | 124 x 125      
32582        | 19100           |                
32825        | 10100           | 100 x 101      
32852        | 19100           |                
35228        | 19100           |                
35282        | 10100           | 100 x 101      
35822        | 15500           | 124 x 125      
38225        | 19100           |                
38252        | 15500           | 124 x 125      
38522        | 10100           | 100 x 101      
52238        | 10100           | 100 x 101      
52283        | 19100           |                
52328        | 19100           |                
52382        | 15500           | 124 x 125      
52823        | 10100           | 100 x 101      
52832        | 15500           | 124 x 125      
53228        | 15500           | 124 x 125      
53282        | 10100           | 100 x 101      
53822        | 19100           |                
58223        | 15500           | 124 x 125      
58232        | 19100           |                
58322        | 10100           | 100 x 101      
82235        | 15500           | 124 x 125      
82253        | 19100           |                
82325        | 19100           |                
82352        | 10100           | 100 x 101      
82523        | 15500           | 124 x 125      
82532        | 10100           | 100 x 101      
83225        | 10100           | 100 x 101      
83252        | 15500           | 124 x 125      
83522        | 19100           |                
85223        | 10100           | 100 x 101      
85232        | 19100           |                
85322        | 15500           | 124 x 125  

Note that it is only when the determinant is 19100 that it is not pronic since 19100 = 100 x 191. In my post titled Determinants of Circulant Matrices, I listed all numbers up to 40000 with the property that the determinants of their circulant matrices were pronic. The numbers between 28000 and 40000 are:

28235, 28253, 28325, 28327, 28453, 28479, 28532, 28543, 28574, 28619, 28732, 28776, 29054, 29168, 29245, 29254, 29555, 29700, 29748, 30171, 30179, 30566, 30575, 30665, 30900, 31100, 31107, 31134, 31233, 31323, 31332, 31355, 31358, 31385, 31400, 31413, 31422, 31440, 31510, 31637, 31646, 31684, 31763, 31907, 32124, 32133, 32223, 32232, 32241, 32258, 32285, 32287, 32313, 32322, 32331, 32528, 32728, 32825, 32845, 32854, 32960, 33123, 33132, 33141, 33176, 33213, 33222, 33231, 33312, 33321, 33335, 33353, 33515, 33518, 33533, 33569, 33671, 33789, 33815, 33965, 33987, 34100, 34166, 34212, 34258, 34311, 34410, 34582, 34599, 34700, 34861, 35153, 35183, 35248, 35282, 35333, 35482, 35507, 35531, 35606, 35693, 35822, 35831, 35936, 35949, 35996, 35999, 36056, 36065, 36137, 36359, 36395, 36418, 36461, 36506, 36614, 36713, 36920, 36995, 37011, 37055, 37091, 37316, 37361, 37400, 37700, 37799, 37822, 37893, 37938, 37979, 38146, 38153, 38252, 38272, 38379, 38397, 38425, 38522, 38524, 38531, 39495, 39536, 39569, 39599, 39653, 39659, 39738, 39797, 39873, 39900, 39954, 39959, 39977, 39995

Note that with 28235 and its digit permutations there are TWO determinants that satisfy. These are:

• \(10100 = 100 \times 101\)
• \(15500 = 124 \times 125\)

This is generally not the case. Consider 28327 and its digit permutations where only the determinant 14762 = 121 x 122 satisfies (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22378        | 14762           | 121 x 122      
22387        | 11462           |                
22738        | 27962           |                
22783        | 11462           |                
22837        | 27962           |                
22873        | 14762           | 121 x 122      
23278        | 27962           |                
23287        | 27962           |                
23728        | 11462           |                
23782        | 14762           | 121 x 122      
23827        | 14762           | 121 x 122      
23872        | 11462           |                
27238        | 14762           | 121 x 122      
27283        | 14762           | 121 x 122      
27328        | 11462           |                
27382        | 27962           |                
27823        | 27962           |                
27832        | 11462           |                
28237        | 11462           |                
28273        | 11462           |                
28327        | 14762           | 121 x 122      
28372        | 27962           |                
28723        | 27962           |                
28732        | 14762           | 121 x 122      
32278        | 11462           |                
32287        | 14762           | 121 x 122      
32728        | 14762           | 121 x 122      
32782        | 27962           |                
32827        | 11462           |                
32872        | 27962           |                
37228        | 27962           |                
37282        | 11462           |                
37822        | 14762           | 121 x 122      
38227        | 27962           |                
38272        | 14762           | 121 x 122      
38722        | 11462           |                
72238        | 11462           |                
72283        | 27962           |                
72328        | 27962           |                
72382        | 14762           | 121 x 122      
72823        | 11462           |                
72832        | 14762           | 121 x 122      
73228        | 14762           | 121 x 122      
73282        | 11462           |                
73822        | 27962           |                
78223        | 14762           | 121 x 122      
78232        | 27962           |                
78322        | 11462           |                
82237        | 14762           | 121 x 122      
82273        | 27962           |                
82327        | 27962           |                
82372        | 11462           |                
82723        | 14762           | 121 x 122      
82732        | 11462           |                
83227        | 11462           |                
83272        | 14762           | 121 x 122      
83722        | 27962           |                
87223        | 11462           |                
87232        | 27962           |                
87322        | 14762           | 121 x 122   

Some Special Pronic Numbers

While driving I'm always on the lookout for interesting number plates and this morning, while driving, I noticed a number plate with numerals that were a concatenation of two consecutive numbers. I can't remember what the number was but the thought then struck me as to whether there were any pronic numbers that could be constructed from concatenated consecutive numbers. If there were then the number would have the property that it was a product of consecutive numbers as well as a concatenation of consecutive numbers (not necessarily the same numbers however).

It's easy enough to check for (permalink) and in the range up to one hundred million, there are only three numbers that qualify. These are 12, 56 and 6162:

• \(12 = 1 \, || \,  2 =2^2 \times 3 = 3 \times 4 \)
• \(56 = 5 \, || \, 6 =2^3 \times 7=7 \times 8 \)
• \(6162 = 61 \, || \, 62 = 2 \times 3 \times 13 \times 79 = 78 \times 79 \)

This stimulated me to look for some other special pronic numbers. My thoughts turned to numbers that share the same digits as their pronic factors. In the range up to slightly over one hundred million, there are quite a few. They are (permalink):

110, 10100, 11990, 123552, 251502, 1001000, 1019090, 1156700, 2987712, 11232552, 13701102, 25015002, 25255650, 25265702, 25720112, 57010050, 57161160, 100010000

The details are as follows:

• 110 = 10 x 11 with shared digits of 0, 1
• 10100 = 100 x 101 with shared digits of 0, 1
• 11990 = 109 x 110 with shared digits of 0, 1, 9
• 123552 = 351 x 352 with shared digits of 1, 2, 3, 5
• 251502 = 501 x 502 with shared digits of 0, 1, 2, 5
• 1001000 = 1000 x 1001 with shared digits of 0, 1
• 1019090 = 1009 x 1010 with shared digits of 0, 1, 9
• 1156700 = 1075 x 1076 with shared digits of 0, 1, 5, 6, 7
• 2987712 = 1728 x 1729 with shared digits of 1, 2, 7, 8, 9
• 11232552 = 3351 x 3352 with shared digits of 1, 2, 3, 5
• 13701102 = 3701 x 3702 with shared digits of 0, 1, 2, 3, 7
• 25015002 = 5001 x 5002 with shared digits of 0, 1, 2, 5
• 25255650 = 5025 x 5026 with shared digits of 0, 2, 5, 6
• 25265702 = 5026 x 5027 with shared digits of 0, 2, 5, 6, 7
• 25720112 = 5071 x 5072 with shared digits of 0, 1, 2, 5, 7
• 57010050 = 7550 x 7551 with shared digits of 0, 1, 5, 7
• 57161160 = 7560 x 7561 with shared digits of 0, 1, 5, 6, 7
• 100010000 = 10000 x 10001 with shared digits of 0, 1

There's plenty more room for investigation but I'll leave if there for now.

Revisiting Sums and Differences of Two Pronic Numbers

On Monday the 6th of February 2023, I made a post titled Even Numbers as Sum and Differences of Two Pronic Numbers and concluded it by stating the following:

This post is just meant as an initial investigation into this topic and it would be useful to expand the investigation to include all even numbers in the range up to about 40000.

In this post, I'd like to explore the conjecture that all even numbers can be expressed, at least in one way, as the sum or difference of two pronic numbers. In my original post, I showed that not all even numbers could be expressed as a sum of two pronic numbers nor as a difference of two pronic numbers. However, I never took it further and that's what I'd like to do in this post.

What's prompted this post is one of the properties of the number associated with my diurnal age today: 27332. It is one of those rare numbers that can be expressed as a sum of two pronic numbers in six different ways:

• 27332 is the sum of 27060 and 272
• 27332 is the sum of 26732 and 600
• 27332 is the sum of 25440 and 1892
• 27332 is the sum of 21170 and 6162
• 27332 is the sum of 20022 and 7310
• 27332 is the sum of 17030 and 10302

Let's recall that pronic numbers are of the form \(n \times (n+1) \) for \(n \geq 1\). Up to 40,000, these numbers are:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660, 3782, 3906, 4032, 4160, 4290, 4422, 4556, 4692, 4830, 4970, 5112, 5256, 5402, 5550, 5700, 5852, 6006, 6162, 6320, 6480, 6642, 6806, 6972, 7140, 7310, 7482, 7656, 7832, 8010, 8190, 8372, 8556, 8742, 8930, 9120, 9312, 9506, 9702, 9900, 10100, 10302, 10506, 10712, 10920, 11130, 11342, 11556, 11772, 11990, 12210, 12432, 12656, 12882, 13110, 13340, 13572, 13806, 14042, 14280, 14520, 14762, 15006, 15252, 15500, 15750, 16002, 16256, 16512, 16770, 17030, 17292, 17556, 17822, 18090, 18360, 18632, 18906, 19182, 19460, 19740, 20022, 20306, 20592, 20880, 21170, 21462, 21756, 22052, 22350, 22650, 22952, 23256, 23562, 23870, 24180, 24492, 24806, 25122, 25440, 25760, 26082, 26406, 26732, 27060, 27390, 27722, 28056, 28392, 28730, 29070, 29412, 29756, 30102, 30450, 30800, 31152, 31506, 31862, 32220, 32580, 32942, 33306, 33672, 34040, 34410, 34782, 35156, 35532, 35910, 36290, 36672, 37056, 37442, 37830, 38220, 38612, 39006, 39402, 39800

Let's look at numbers in the region of 27332 to get an idea of what's going on (permalink):

• 27334 is the difference of 27390 and 56
• 27334 is the difference of 45156 and 17822
  
  

• 27336 is the sum of 26406 and 930
• 27336 is the difference of 28392 and 1056
• 27336 is the difference of 56406 and 29070
• 27336 is the difference of 86142 and 58806
• 27336 is the difference of 338142 and 310806
• 27336 is the difference of 660156 and 632820
• 27336 is the difference of 2932656 and 2905320
  
  

• 27338 is the difference of 46724060 and 46696722
  
  

• 27340 is the difference of 480942 and 453602
  
  

• 27342 is the difference of 27722 and 380
• 27342 is the sum of 26082 and 1260
• 27342 is the difference of 29412 and 2070
• 27342 is the sum of 22650 and 4692
• 27342 is the difference of 35532 and 8190
• 27342 is the difference of 37442 and 10100
• 27342 is the difference of 61752 and 34410
• 27342 is the difference of 63252 and 35910
• 27342 is the difference of 92112 and 64770
• 27342 is the difference of 120062 and 92720
• 27342 is the difference of 208392 and 181050
• 27342 is the difference of 437582 and 410240
• 27342 is the difference of 590592 and 563250
• 27342 is the difference of 967272 and 939930
• 27342 is the difference of 2321052 and 2293710
• 27342 is the difference of 3827892 and 3800550
  
  

• 27344 is the sum of 23562 and 3782
• 27344 is the sum of 16002 and 11342
• 27344 is the difference of 743906 and 716562
  
  

• 27346 is the difference of 30102 and 2756
• 27346 is the difference of 31506 and 4160
• 27346 is the difference of 400056 and 372710
• 27346 is the difference of 1558752 and 1531406
  
  

• 27348 is the difference of 27390 and 42
• 27348 is the difference of 29070 and 1722
• 27348 is the difference of 80940 and 53592
• 27348 is the difference of 115260 and 87912
• 27348 is the difference of 1312170 and 1284822

At first I couldn't find a sum or difference for 27338 until I extended the list of pronic numbers to 10,000 x 10,001. My initial analysis suggests that it is possible to write any even number as the sum or difference of two pronic numbers. Not surprisingly differences seem to predominate.

At first I thought that 27322 was anomalous in that it is the sum of six different pairs of pronic numbers but no differences. However, extending the list of pronic numbers as before I found that it is also the difference of 11686142 and 11658810. There may well be a formal mathematical proof of my conjecture but my approach so far is purely empirical. Put in formal mathematical terms, I could state my conjecture as follows:

For any even positive integer \(n \gt 2\), there exists pronic numbers \(p \times (p+1) \) and \(q \times (q+1) \) with \(p \gt q\) such that:$$ \begin{align} n &=p \times (p+1) +q \times (q+1) \\ &=p^2+q^2 +p+q \end{align}$$or$$ \begin{align} n &=p \times (p+1) -q \times (q+1) \\ &= p^2-q^2+p-q \\ &= (p+q)(p-q) + p-q \\ &= (p+q)(p-q+1) \end{align}$$

As a final note, let's not forget that every pronic number is twice a triangular number since the formula for a triangular \(T_n\) number is:$$ \begin{align} T_n &= \sum_{k=1}^n k \\ &= 1+2+3 \dots n \\ &=\frac{n \times (n+1)}{2} \\&= \binom{n}{2} \end{align} $$Thus many of the properties of triangular numbers would be shared by pronic numbers.

Midway Between Cubics and Pronics

Recently I posted about distances to cubic numbers (In the Vicinity of Cubic Numbers) as well as the product of three consecutive integers (Infinite Sums of Reciprocals of Pronic Numbers). These latter numbers are referred to variously as pronic, promic and oblong numbers. The number associated with my diurnal age today, 26985, involves both cubic and pronic numbers and qualifies it for inclusion in OEIS A342873:

A342873

Numbers whose distance to the nearest cube equals the distance to the nearest product of 3 consecutive integers (three-dimensional oblong).

My approach to generating the terms of this sequence, using SageMath, was to first generate, separately, the sequence of cubic numbers and the sequence of oblong numbers up to a little over 42,000. 

The sequence of 36 cubic numbers (including zero) is:

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875

The sequence of 36 oblong numbers (including zero) is:

0, 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, 2184, 2730, 3360, 4080, 4896, 5814, 6840, 7980, 9240, 10626, 12144, 13800, 15600, 17550, 19656, 21924, 24360, 26970, 29760, 32736, 35904, 39270, 42840, 46620

Combined these two sets of numbers together gives a total of 71 numbers since zero is duplicated:

0, 1, 6, 8, 24, 27, 60, 64, 120, 125, 210, 216, 336, 343, 504, 512, 720, 729, 990, 1000, 1320, 1331, 1716, 1728, 2184, 2197, 2730, 2744, 3360, 3375, 4080, 4096, 4896, 4913, 5814, 5832, 6840, 6859, 7980, 8000, 9240, 9261, 10626, 10648, 12144, 12167, 13800, 13824, 15600, 15625, 17550, 17576, 19656, 19683, 21924, 21952, 24360, 24389, 26970, 27000, 29760, 29791, 32736, 32768, 35904, 35937, 39270, 39304, 42840, 42875, 46620

Fortunately the order of these numbers, after zero, alternates from cubic to oblong and this algorithm was able to be applied in order to identify the numbers that satisfied the criterion imposed by OEIS A342873. These are the resulting numbers up to a little over 40,000:

0, 7, 16, 62, 92, 213, 276, 508, 616, 995, 1160, 1722, 1956, 2737, 3052, 4088, 4496, 5823, 6336, 7990, 8620, 10637, 11396, 13812, 14712, 17563, 18616, 21938, 23156, 26985, 28380, 32752, 34336, 39287, 41072

For example, today's number of 26985 is a distance of 15 from the nearest cubic number (27000 = 30 x 30 x 30) and the same distance from the nearest oblong numbers (26970 = 29 x 30 x 31). 

Had the two sets of numbers become jumbled up when combined, the task of identifying suitable numbers would have been more difficult. However, the oblong numbers \(n \times (n+1) \times (n+2) \) are only a little ahead of corresponding cubic numbers (\(n^3 )\) and so the problem doesn't arise.

To see that the oblong number following the cube is always less than the next cube, consider the following:$$ \begin{align} n  (n+1) (n+2) &=n^3 + 3n^2 + 2n\\(n+1)^3&=n^3+3n^2+3n+1 \end{align}$$Clearly the next cubic number is always \(n+1\) ahead of the oblong number. The same reasoning would apply if we looked at numbers that are equidistant from the nearest square number and the nearest pronic number.

The earlier algorithm is easily modified to produce these numbers that constitute OEIS A074378:

A074378

Numbers whose distance to nearest square number equals their distance to nearest pronic number.

The modified algorithm generates these numbers:

0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333, 390, 410, 473, 495, 564, 588, 663, 689, 770, 798, 885, 915, 1008, 1040, 1139, 1173, 1278, 1314, 1425, 1463, 1580, 1620, 1743, 1785, 1914, 1958, 2093, 2139, 2280, 2328, 2475, 2525, 2678, 2730, 2889, 2943, 3108, 3164, 3335, 3393, 3570, 3630, 3813, 3875, 4064, 4128, 4323, 4389, 4590, 4658, 4865, 4935, 5148, 5220, 5439, 5513, 5738, 5814, 6045, 6123, 6360, 6440, 6683, 6765, 7014, 7098, 7353, 7439, 7700, 7788, 8055, 8145, 8418, 8510, 8789, 8883, 9168, 9264, 9555, 9653, 9950, 10050, 10353, 10455, 10764, 10868, 11183, 11289, 11610, 11718, 12045, 12155, 12488, 12600, 12939, 13053, 13398, 13514, 13865, 13983, 14340, 14460, 14823, 14945, 15314, 15438, 15813, 15939, 16320, 16448, 16835, 16965, 17358, 17490, 17889, 18023, 18428, 18564, 18975, 19113, 19530, 19670, 20093, 20235, 20664, 20808, 21243, 21389, 21830, 21978, 22425, 22575, 23028, 23180, 23639, 23793, 24258, 24414, 24885, 25043, 25520, 25680, 26163, 26325, 26814, 26978, 27473, 27639, 28140, 28308, 28815, 28985, 29498, 29670, 30189, 30363, 30888, 31064, 31595, 31773, 32310, 32490, 33033, 33215, 33764, 33948, 34503, 34689, 35250, 35438, 36005, 36195, 36768, 36960, 37539, 37733, 38318, 38514, 39105, 39303, 39900, 40100

For example, the number 14 in this sequence is an equal distance from 12 = 3 x 4 and 16 = 4 x 4. The algorithm could be extended (permalink) the other way to find numbers that are equidistant from the nearest fourth power and the number that is a product of four consecutive integers. The initial resultant numbers are not a part of any OEIS sequence but they are as follows:

0, 20, 188, 308, 1068, 1488, 3560, 4568, 8960, 10940, 18948, 22380, 35588, 41048, 61328, 69488, 99000, 110628, 151820, 167780

For example the number 20 is equidistant from 16 = 2 x 2 x 2 x 2 and 24 = 1 x 2 x 3 x 4. The algorithm could be extended indefinitely but to little purpose. Nonetheless, it's been an interesting exercise.

Even Numbers as Sums and Differences of Two Pronic Numbers

It was a form of Goldbach's conjecture that got me thinking about an equivalent involving pronic numbers. The conjecture is that every even number can be written as a sum of two prime numbers. Pronic numbers, that is numbers of the form \(n \times (n-1) \), are always even and so the sum of two pronic numbers must be even as well. 

Can every even number be written as a sum of two not necessarily distinct pronic numbers. Let's consider the numbers from 1 to 100. The results are as follows:

• 4 = 2 + 2
• 8 = 2 + 6
• 12 = 6 + 6
• 14 = 2 + 12
• 18 = 6 + 12
• 22 = 2 + 20
• 24 = 12 + 12
• 26 = 6 + 20
• 32 = 2 + 30
• 32 = 12 + 20
• 36 = 6 + 30
• 40 = 20 + 20
• 42 = 12 + 30
• 44 = 2 + 42
• 48 = 6 + 42
• 50 = 20 + 30
• 54 = 12 + 42
• 58 = 2 + 56
• 60 = 30 + 30
• 62 = 6 + 56
• 62 = 20 + 42
• 68 = 12 + 56
• 72 = 30 + 42
• 74 = 2 + 72
• 76 = 20 + 56
• 78 = 6 + 72
• 84 = 12 + 72
• 84 = 42 + 42
• 86 = 30 + 56
• 92 = 2 + 90
• 92 = 20 + 72
• 96 = 6 + 90
• 98 = 42 + 56

As can be seen, not all even numbers between 4 and 100 can be written as a sum of two pronic numbers. We are missing 6, 10, 16, 20, 28, 30, 34, 38, 46, 52, 56, 64, 70, 80, 82, 88, 90, 94 and 100. Some numbers however, can be written as a sum of two pronic numbers in more than one way. These numbers are 32, 62, 84 and 92. Each can be written in two ways.

Let's take another range of one hundred numbers, this time from 26900 to 27000. The results are as follows:

• 26906 = 4556 + 22350
• 26912 = 506 + 26406
• 26912 = 6320 + 20592
• 26912 = 13340 + 13572
• 26914 = 182 + 26732
• 26916 = 3660 + 23256
• 26916 = 8010 + 18906
• 26916 = 8556 + 18360
• 26916 = 13110 + 13806
• 26922 = 1482 + 25440
• 26922 = 10920 + 16002
• 26924 = 12882 + 14042
• 26928 = 1806 + 25122
• 26930 = 9900 + 17030
• 26936 = 2756 + 24180
• 26936 = 12656 + 14280
• 26940 = 4290 + 22650
• 26942 = 210 + 26732
• 26942 = 2450 + 24492
• 26942 = 7482 + 19460
• 26942 = 9120 + 17822
• 26948 = 6642 + 20306
• 26950 = 1190 + 25760
• 26950 = 3080 + 23870
• 26952 = 870 + 26082
• 26952 = 12432 + 14520
• 26958 = 552 + 26406
• 26968 = 2162 + 24806
• 26968 = 10712 + 16256
• 26972 = 240 + 26732
• 26972 = 12210 + 14762
• 26984 = 3422 + 23562
• 26984 = 4032 + 22952
• 26994 = 6972 + 20022
• 26994 = 9702 + 17292
• 26996 = 11990 + 15006
• 27000 = 1560 + 25440

There are 21 numbers in the range of 51 even numbers that can be expressed as a sum of two pronic numbers. This is a percentage of about 42%. These are (permalink):

26906, 26912, 26912, 26912, 26914, 26916, 26916, 26916, 26916, 26922, 26922, 26924, 26928, 26930, 26936, 26936, 26940, 26942, 26942, 26942, 26942, 26948, 26950, 26950, 26952, 26952, 26958, 26968, 26968, 26972, 26972, 26984, 26984, 26994, 26994, 26996, 27000

Some numbers can be expressed as a sum in four ways. Here is the count of ways with four ways numbers marked in bold): 

(26906, 1), (26912, 3), (26914, 1), (26916, 4), (26922, 2), (26924, 1), (26928, 1), (26930, 1), (26936, 2), (26940, 1), (26942, 4), (26948, 1), (26950, 2), (26952, 2), (26958, 1), (26968, 2), (26972, 2), (26984, 2), (26994, 2), (26996, 1), (27000, 1)

Thus is the range selected there are only two numbers that meet the four ways criterion and these are 26916 and 26942. This is a percentage of about 4%. What becomes of interest then is what numbers in a given range can be expressed as a sum of two pronic numbers in exactly four ways. Let's consider a range of two thousand, from 26000 to 28000. There are only 33 such numbers, representing about 3.3% of the range, and they are (permalink):

26032, 26052, 26102, 26214, 26222, 26292, 26312, 26342, 26382, 26432, 26630, 26682, 26752, 26868, 26916, 26942, 27042, 27062, 27072, 27102, 27176, 27192, 27242, 27402, 27480, 27522, 27572, 27602, 27752, 27852, 27854, 27922, 27990

Some numbers can be written is five different ways. There are two numbers in the range between 26000 and 28000 and they are 26732 and 27812. Other numbers can be written as a sum of pronic numbers in six different ways. These numbers are 26462, 26562, 27162, 27332 and 27462. Below is shown the sums for 27462:

• 27462 = 72 + 27390
• 27462 = 1056 + 26406
• 27462 = 2970 + 24492
• 27462 = 5112 + 22350
• 27462 = 8556 + 18906
• 27462 = 12210 + 15252

There are no numbers in the range that can be expressed as a sum in seven ways. Notice that throughout this investigation, we imposed the condition that the two pronic numbers need not be distinct. If we imposed the condition that they must be distinct, then there would be slightly fewer numbers that satisfy.

It's also possible to consider numbers that are a difference of two pronic numbers. For example in the range between 26900 and 27000, there are 35 numbers out the 51 even numbers that can expressed as differences, many in multiple ways. However, there are 16 that cannot and these are (permalink):

26902, 26914, 26916, 26926, 26930, 26932, 26938, 26948, 26954, 26958, 26968, 26974, 26982, 26984, 26990, 26998

This post is just meant as an initial investigation into this topic and it would be useful to expand the investigation to include all even numbers in the range up to about 40000.

Infinite Sums of Reciprocals of Pronic Numbers

I'd heard the term pronic number before but not promic number. However, the number associated with my diurnal age today, 26970, is such a number according to OEIS  A007531:

A007531

\(a(n) = n \times (n-1) \times (n-2)   \)

The sequence runs:

0, 0, 0, 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, 2184, 2730, 3360, 4080, 4896, 5814, 6840, 7980, 9240, 10626, 12144, 13800, 15600, 17550, 19656, 21924, 24360, 26970, 29760, 32736, 35904, 39270, 42840, 46620, 50616, 54834, 59280, 63960, 68880

Interestingly, the next such number, 29760, is a permutation of the digits of 26970. The comments to OEIS A007531 state that members of this sequence are "three-dimensional promic (or oblong) numbers". While I initially thought that I hadn't heard the term "promic" before, I realised in fact that I had and it was in the entry for Pronic Number is Wolfram MathWorld. Here is the text of interest:

Pronic numbers are also known as oblong (Merzbach and Boyer 1991, p. 50) or heteromecic numbers. However, "pronic" seems to be a misspelling of "promic" (from the Greek promekes, meaning rectangular, oblate, or oblong). However, no less an authority than Euler himself used the term "pronic," so attempting to "correct" it at this late date seems inadvisable.

So that clears matters up and it would seem that the terms pronic and oblong can be applied to numbers of the form \(n \times (n-1) \) with \(n \geq 1\) as well as those of the form \(n \times (n-1) \times (n-2) \) with \(n \geq 2\). Anyway, this is just nomenclature and these types of numbers might indeed be referred to as generalised factorial numbers because they can be written in the following form:$$P_{n,k}=\frac{n!}{(n-k)!} \text{ where } k \geq 2$$The notation \(P_{n,k}\) can be interpreted as the \(n\)-th pronic number of dimension \(k\).  When \(k=2\) the dimensionality is two, when \(k=3\) the dimensionality is three and so on. In this notational system, we have:$$26970=P_{31,3}=\frac{31!}{(31-3)!}=31 \times 30 \times 29$$The sequences for various values of \(k\) can be generated using this SageMath code. For example, when \(k=4\), the sequence becomes OEIS A052762:

24, 120, 360, 840, 1680, 3024, 5040, 7920, 11880, 17160, 24024, 32760, 43680, 57120, 73440, 93024, 116280, 143640, 175560, 212520, 255024, 303600, 358800, 421200, 491400, 570024, 657720, 755160, 863040, 982080, 1113024, 1256640, 1413720, 1585080, 1771560, 1974024, 2193360, 2430480, 2686320, 2961840, 3258024, 3575880, 3916440, 4280760, 4669920, 5085024, 5527200

In my post titled My Yearly Pronic Number on Saturday, 11th June 2022, I noted that:$$\sum_{n=1}^{\infty} \frac{1}{n \times (n+1)}=1$$To this it can be added that:

$$\sum_{n=1}^{\infty} \frac{1}{n \times (n+1) \times (n+2)}=\frac{1}{4}$$The infinite sums can be worked out for any value of \(k\) in \(P_{n,k} \) using this permalink. For \(k=4\), the value is 1/18. To summarise, the infinite sums are:

• \( \displaystyle \sum_{n=2}^{\infty} \dfrac{1}{P_{n,2}}=1 \)
• \( \displaystyle \sum_{n=3}^{\infty} \dfrac{1}{P_{n,3}}=\dfrac{1}{4} \)
• \( \displaystyle \sum_{n=4}^{\infty} \dfrac{1}{P_{n,4}}=\dfrac{1}{18} \)
• \( \displaystyle \sum_{n=5}^{\infty} \dfrac{1}{P_{n,5}}=\dfrac{1}{96} \)
• \( \displaystyle \sum_{n=6}^{\infty} \dfrac{1}{P_{n,6}}=\dfrac{1}{600} \)

In general, the result is that:$$ \sum_{n=k}^{\infty} \dfrac{1}{P_{n,k}}=\dfrac{1}{k!-(k-1)!}  \text{ for } k \geq 2$$

My Yearly Pronic Number

Pronic numbers are numbers of the form \(n \times (n+1) \) where \(n\) is an integer \( \geq 1\). Thus the first such number is 2. Here are the pronic numbers up to 40,000:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660, 3782, 3906, 4032, 4160, 4290, 4422, 4556, 4692, 4830, 4970, 5112, 5256, 5402, 5550, 5700, 5852, 6006, 6162, 6320, 6480, 6642, 6806, 6972, 7140, 7310, 7482, 7656, 7832, 8010, 8190, 8372, 8556, 8742, 8930, 9120, 9312, 9506, 9702, 9900, 10100, 10302, 10506, 10712, 10920, 11130, 11342, 11556, 11772, 11990, 12210, 12432, 12656, 12882, 13110, 13340, 13572, 13806, 14042, 14280, 14520, 14762, 15006, 15252, 15500, 15750, 16002, 16256, 16512, 16770, 17030, 17292, 17556, 17822, 18090, 18360, 18632, 18906, 19182, 19460, 19740, 20022, 20306, 20592, 20880, 21170, 21462, 21756, 22052, 22350, 22650, 22952, 23256, 23562, 23870, 24180, 24492, 24806, 25122, 25440, 25760, 26082, 26406, 26732, 27060, 27390, 27722, 28056, 28392, 28730, 29070, 29412, 29756, 30102, 30450, 30800, 31152, 31506, 31862, 32220, 32580, 32942, 33306, 33672, 34040, 34410, 34782, 35156, 35532, 35910, 36290, 36672, 37056, 37442, 37830, 38220, 38612, 39006, 39402, 39800

I've marked the pronic number 26732 = 163 x 164 in bold because that is my diurnal age today (June 11th 2022) and this fact is what prompted me to make this post. The previous such number (26406 = 162 x 163) occurred on Tuesday, July 20th 2021 and the next (27060 = 164 x 165) will occur on Friday, May 5th 2023. So at the moment, a pronic number appearing as my diurnal age is pretty much a yearly thing and as such should be celebrated.

Pronic numbers are also called oblong numbers, rectangular numbers or heteromecic numbers. Interestingly, the sum of the reciprocals of the pronic numbers is 1. Thus:$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=1$$I've written about numbers of this sort before in a post titled Pronic Pandigital Numbers and Beyond on July 23rd 2021. Over 80% of pronic numbers are abundant but 26732 is deficient. In fact, of the 199 numbers in the list above, only 35 are deficient. These are:

2, 110, 182, 506, 1406, 1892, 2162, 2756, 3422, 3782, 4556, 5402, 6806, 7310, 8930, 9506, 11342, 11990, 14042, 14762, 17030, 17822, 18632, 20306, 21170, 22052, 22952, 24806, 26732, 27722, 29756, 31862, 32942, 36290, 37442

This sequence of numbers forms part of OEIS A077804:

A077804

Deficient oblong numbers.

The generating function for the pronic numbers is:$$\frac{2x}{(1-x)^3}=2x+6x^2+12x^3+20x^4+ \dots$$Pronic numbers are also figurate numbers of the form:$$P_n=2T_n=n(n+1)$$where \(T_n\) is the \(n^{th}\) triangular number. A very few pronic numbers are palindromic. The first few are listed below:

2, 6, 272, 6006, 289982, 2629262, 6039306, 27999972, 28233282, 2704884072, 20278187202, 20591819502, 2592587852952, 2936231326392, 21809166190812, 27237788773272, 229145919541922, 233552101255332, 250087292780052, 2243922442293422, 2570769009670752, 20333113431133302, 27785925652958772

These numbers form OEIS A028337:

A028337

Palindromes of the form n(n+1).