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Oblong numbers
From OeisWiki
Oblong numbers are two dimensional figurate numbers of the form
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.
Contents |
Formulae
Halfway between any two consecutive oblong numbers, one finds the square
where is the ^{ th} square.
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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 |
- {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
Harmonic series of the oblong numbers
The harmonic series of the oblong numbers (sum of reciprocals of oblong numbers) is
Almost-oblong numbers
A028387 Almost-oblong numbers: .
- {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: . (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, ...}