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).