Oblong numbers

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Oblong numbers are two dimensional figurate numbers of the form

on  =  n (n + 1)  =  2 tn , n ≥ 0,

where

tn

is the

n

th triangular number.
A002378 Oblong (or promic, pronic, or heteromecic) numbers:

n (n + 1), n   ≥   0

.

{0, 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, ...}

Oblong numbers, being the product of two consecutive integers, are all even (twice a triangular number) and obviously composite when greater than 2.

Formulae

n																																																																		$\textstyle {\sum _{i=o_{n-1}}^{o_{n}}i=}$ n 2 (2 n + 1) A099721
0																																	0																																	0
1																															0		1		2																															3
2																													2		3		4		5		6																													20
3																											6		7		8		9		10		11		12																											63
4																									12		13		14		15		16		17		18		19		20																									144
5																							20		21		22		23		24		25		26		27		28		29		30																							275
6																					30		31		32		33		34		35		36		37		38		39		40		41		42																					468

Halfway between any two consecutive oblong numbers, one finds the square

sn  ≡  n 2  =  on + on  − 1 2 , n ≥ 1,

where

sn

is the

n

th square.
A099721

n 2 (2 n + 1), n   ≥   0.

$\textstyle {\left(\sum _{i{\;\;\!\!\!}={\;\;\!\!\!}o_{n-1}}^{o_{n}}i,\,n\geq 1.\right)}$

{0, 3, 20, 63, 144, 275, 468, 735, 1088, 1539, 2100, 2783, 3600, 4563, 5684, 6975, 8448, 10115, 11988, 14079, 16400, 18963, 21780, 24863, 28224, 31875, 35828, 40095, ...}

Recurrence

     ${\begin{array}{l}\displaystyle {o_{n}={\begin{cases}0,&{\text{if }}n=0;\\o_{n-1}+2n,&{\text{if }}n\geq 1.\end{cases}}}\end{array}}$

Generating function

G{on, n   ≥  0}(x)  ≡    ∞ ∑   n   = 0    on x n  =  2 x (1 − x) 3  .

Harmonic series of the oblong numbers

The harmonic series of the oblong numbers (sum of reciprocals of oblong numbers) converges to 1, since

∞ ∑   n  = 1       1 n (n + 1)  =    ∞ ∑   n  = 1    1 n  − 1 n + 1  =

∞ ∑   n  = 1    1 n  −      ∞ ∑   n  = 2    1 n   =  1.

Almost-oblong numbers

A028387 Almost-oblong numbers:

on  −  1 = n (n + 1)  −  1, n   ≥   1

.

{1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, ...}

Almost-oblong primes

A002327 Almost-oblong primes: primes of form

n 2  −  n  −  1

.

{5, 11, 19, 29, 41, 71, 89, 109, 131, 181, 239, 271, 379, 419, 461, 599, 701, 811, 929, 991, 1259, 1481, 1559, 1721, 1979, 2069, 2161, 2351, 2549, 2861, 2969, 3079, 3191, ...}

It is conjectured that there is an infinity of primes of this form, although it is not proved yet.

A?????? Almost-oblong composites: composites of form

n 2  −  n  −  1

.

{55, 155, 209, 305, 341, 505, 551, 649, 755, 869, 1055, 1121, 1189, 1331, ...}

Quasi-oblong numbers

A002061 Quasi-oblong numbers:

on  − 1 + 1 = n 2  −  n + 1, n   ≥   1

. (Central polygonal numbers.)

{1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, ...}

Quasi-oblong primes

A002383 Quasi-oblong primes: primes of form

n 2 + n + 1

.

{3, 7, 13, 31, 43, 73, 157, 211, 241, 307, 421, 463, 601, 757, 1123, 1483, 1723, 2551, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, ...}

It is conjectured that there is an infinity of primes of this form, although it is not proved yet.

A174969 Quasi-oblong composites: composites of form

n 2 + n + 1

.

{21, 57, 91, 111, 133, 183, 273, 343, 381, 507, 553, 651, 703, 813, 871, 931, 993, 1057, 1191, 1261, 1333, 1407, 1561, 1641, 1807, 1893, 1981, 2071, 2163, 2257, 2353, ...}

See also

• Triangular numbers
• Square numbers