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Oblong numbers are two dimensional figurate numbers of the form
-
on = n (n + 1) = 2 tn , n ≥ 0, |
where
is the
th triangular number.
A002378 Oblong (or
promic,
pronic, or
heteromecic) numbers:
.
-
{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
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A099721
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0
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0
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0
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1
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0
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1
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2
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3
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2
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2
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3
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4
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5
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6
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20
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3
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6
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7
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8
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9
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10
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11
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12
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63
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4
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12
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13
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14
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15
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16
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17
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18
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19
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20
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144
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5
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20
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21
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22
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23
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24
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25
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26
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27
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28
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29
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30
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275
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6
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30
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31
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32
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33
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34
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35
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36
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37
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38
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39
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40
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41
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42
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468
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Halfway between any two consecutive oblong numbers, one finds the square
-
where
is the
th square.
A099721
-
{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
Generating function
-
G{on, n ≥ 0}(x) ≡ on x n = . |
Harmonic series of the oblong numbers
The harmonic series of the oblong numbers (sum of reciprocals of oblong numbers) converges to 1, since
-
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
.
-
{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
.
-
{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
.
-
{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
.
-
{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