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).