# [seqfan] Re: Numbers of the form n*(n+k)

israel at math.ubc.ca israel at math.ubc.ca
Thu Mar 19 19:52:50 CET 2020

```The terminologies "oblong number", "rectangular number", "heteromecic
number", "pronic number" and "promic number" are rather a mess, but we're
stuck with them because they've been used historically. Strangely enough,
the quotation for "oblong number" cited by Wiktionary
<https://en.wiktionary.org/wiki/oblong_number> seems to indicate that for
Nicomachus the oblong numbers were specifically **not** what we would call
oblong numbers:

1952 [c. 100], Nicomachus of Gerasa, Martin Luther D'Ooge, transl.;
Robert Maynard Hutchins, Mortimer J. Adler, Wallace Brockway, editors,
Introduction to Arithmetic II (Great Books Of The Western World; 11),
William Benton (Encyclopædia Britannica, Inc.), page 838:

If, however, the sides differ otherwise than by 1, for instance, by
2, 3, 4 or succeeding numbers, as in 2 times 4, 3 times 6, 4 times 8, or
however else they may differ, then no longer will such a number be properly
called a heteromecic, but an oblong number.

Then there's "pronic" as a misspelling of "promic", but if Euler used it,
who are we to object?

Cheers,
Robert

On Mar 19 2020, David Seal wrote:

> I agree with Neil - though since I'm fairly certain I've never come
> across "oblong number" before in the ~55 years since I first encountered
> "square number" and "triangular number", I think there's a good case to
> be made that the old law has been broken in the past!
>
> And I would add another law that "if new terminology is needed, thou
> shalt modify related existing terminology in preference to inventing
> entirely new terminology". For instance, various modifications of the
> idea of "perfect number" have been created, and their names are
> modifications of the term, such as "k-perfect number", "k-imperfect
> number", "semiperfect number", "hemiperfect number", "hyperperfect
> number", "superperfect number", etc.
>
> So if you really do feel that you need terminology for this concept, I'd
> suggest basing it on the existing "oblong number", e.g. by using
> "k-oblong number" for numbers of the form n(n+k), rather than on an
> essentially new term such as "product number".
>
>David
>
>
>> On 19 March 2020 at 14:39 Neil Sloane <njasloane at gmail.com> wrote:
>>
>>
>> Don't think that is a good idea. There is an old law that "thou shalt
>> not create new terminology unnecessarily".
>>
>> Let's stick to n(n+2).
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Email: njasloane at gmail.com
>>
>>
>>
>> On Thu, Mar 19, 2020 at 10:33 AM MARION <charliemath at optonline.net>
>> wrote:
>>
>> > Dear SeqFans,
>> >
>> > I have a question regarding terminology.
>> >
>> > We call the terms of A000290   0, 1, 4, 9, ...  the (perfect) squares.
>> >
>> > We call the terms of A002378   0, 2, 6, 12 ... the oblongs.
>> >
>> > What do we call the terms of A005563  0, 3, 8, 15?
>> >
>> > What do we call the terms of A028552  0, 4, 10, 18?
>> >
>> > What do you think about calling them the +2products and +3products,
>> > respectively? Thus, the squares would be the +0products and the
>> > oblongs, the +1products. Note that I'm not advocating changing the way
>> > we refer to the squares or the oblongs. I'm simply looking for another
>> > way to refer to the terms in sequences like A005563 and A028552.
>> >
>> > I would like to call them something other than "the numbers of the
>> > form n*(n+2)" or "the numbers of the form n*(n+3)." Perhaps I'm just
>> > not aware of some other "shortcut."
>> >
>> > Thanks for any feedback,
>> >
>> > Charlie Marion
>> >
>> > Yorktown Heights New York
>> >
>> >
>> > --
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>
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>

```