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×(n−1)×(n−2) |
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×(n−1) with n≥1 as well as those of the form n×(n−1)×(n−2) with n≥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:Pn,k=n!(n−k)! where k≥2
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:∞∑n=11n×(n+1)=1
- ∞∑n=21Pn,2=1
- ∞∑n=31Pn,3=14
- ∞∑n=41Pn,4=118
- ∞∑n=51Pn,5=196
- ∞∑n=61Pn,6=1600
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