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Oblong numbers

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Oblong numbers are two dimensional figurate numbers of the form

o_{n} = n \, (n+1) = 2 \, t_{n},\quad n \ge 0, \,

where \scriptstyle t_{n} \, is the \scriptstyle n \,th triangular number.

A002378 Oblong (or promic, pronic, or heteromecic) numbers: \scriptstyle n \, (n+1),\, n \,\ge\, 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.

Contents

Formulae

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

s_{n} \equiv n^2 = \frac{o_{n} + o_{n-1}}{2},\quad n \ge 1, \,

where \scriptstyle s_{n} \, is the \scriptstyle n \,th square.

\scriptstyle n \, \scriptstyle \sum_{i=o_{n-1}}^{o_{n}} i \,=\, \,

\scriptstyle n^2 \, (2n+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


A099721 \scriptstyle n^2 \, (2n+1),\, \,n \ge\, 0. \ \left( \sum_{i=o_{n-1}}^{o_{n}} i,\, n \,\ge\, 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

o_{0} = 0; \,
o_{n} = o_{n-1} + 2n,\quad n \ge 1. \,

Generating function

G_{\{ o_{n},\, n \,\ge\, 0 \}}(x) \equiv \sum_{n=0}^{\infty} o_{n} \, x^n = \frac{2 x}{(1-x)^3}. \,

Harmonic series of the oblong numbers

The harmonic series of the oblong numbers (sum of reciprocals of oblong numbers) is

\sum_{n=1}^{\infty} \frac{1}{n \, (n+1)} = \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right) = \left( \sum_{n=1}^{\infty} \frac{1}{n} \right) - \left( \sum_{n=2}^{\infty} \frac{1}{n} \right) = 1. \,

Almost-oblong numbers

A028387 Almost-oblong numbers: \scriptstyle o_{n} - 1 \,=\, n \, (n+1) - 1,\, n \,\ge\, 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 \scriptstyle 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 \scriptstyle 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: \scriptstyle o_{n-1} + 1 \,=\, n^2 - n + 1,\, n \,\ge\, 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 \scriptstyle 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 \scriptstyle 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

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