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# Index to OEIS: Section O

# Index to OEIS: Section O

- This is a section of the Index to the
**OEIS®**. - For further information see the main
**Index to OEIS**page. - Please read Index: Instructions For Updating Index to OEIS before making changes to this page.
- If you did not find what you were looking for in this Index, you can always search the database for a particular word or phrase.
- Full list of sections:

[ Aa | Ab | Al | Am | Ap | Ar | Ba | Be | Bi | Bl | Bo | Br | Ca | Ce | Ch | Cl | Coa | Coi | Com | Con | Cor | Cu | Cy | Da | De | Di | Do | Ea | Ed | El | Eu | Fa | Fe | Fi | Fo | Fu | Ga | Ge | Go | Gra | Gre | Ha | He | Ho | Ia | In | J | K | La | Lc | Li | Lo | Lu | M | Mag | Map | Mat | Me | Mo | Mu | N | Na | Ne | Ni | No | Nu | O | Pac | Par | Pas | Pea | Per | Ph | Poi | Pol | Pos | Pow | Pra | Pri | Pro | Ps | Qua | Que | Ra | Rea | Rel | Res | Ro | Ru | Sa | Se | Si | Sk | So | Sp | Sq | St | Su | Sw | Ta | Te | Th | To | Tra | Tri | Tu | U | V | Wa | We | Wi | X | Y | Z | 1 | 2 | 3 | 4 ]

O'Nan group: A003919, A008625

obtaining numbers from other numbers and the operations of addition, subtraction, etc: see under four 4's problem

octagonal numbers: A000567*

octahedral numbers: A005900*

octahedron, truncated: see truncated octahedron

octahedron: A005899

** octal numbers, sequences related to : **

octupi: A029767*

** odd numbers , sequences related to : **

- odd numbers n such that 2^k + n is composite for all k: see A076336
- odd numbers, fake: A080591
- odd numbers: A005408*
- odd numbers: see also A000700, A000069, A007697, A006046, A007455, A007482, A000593, A007483, A006945, A001033, A002309, A006285, A002594, A006038

** odd perfect numbers , sequences where such numbers (if they exist at all) must occur in : **

- odd perfect numbers (must occur in): sequences for which no odd terms > 1 are currently known: A000396*, A326051*, A001599, A007691, A294900, A324643, A325637, A325638, A325639, A325812, A326131, A326145
- odd perfect numbers (must occur in): sequences that satisfy Euler's criterion and its further restrictions: A228058*, A228059, A326137*, A325376, A325380, A325822
- odd perfect numbers (must occur in): sequences that satisfy some bitwise AND/OR condition: A324643, A324647, A324718, A324719, A324727, A324897 (A324898)
- odd perfect numbers (must occur in): sequences that satisfy some gcd(sigma(n)-X-n, n-X) condition: A007691, A325637, A325812, A325979, A325981, A326063, A326064, A326074, A326131, A326134, A326141, A326145, A326148
- odd perfect numbers (must occur in): sequences that satisfy some other condition: A326051*, A019283, A326181, A005835, A023196, A263837, A162284, A046311, A294900, A325808, A325638, A325639, A326138

odd unimodular lattices, see: lattices, unimodular

odious numbers: A000069*

** Olympiads and other Mathematical competitions, sequences related to : **

- Austria (1985 - Final round - Pb. 2): a(n) = Sum_{k = 1..n} (n – k + 1)^k ==> Min_{n >= 1} a(n+1)/a(n) = 8/3: A003101
- British (BMO - 1976 - Round 1, Pb. 4): a(n) = 19 * 8^n + 17 for n >= 0 is never a prime number: A330770
- British (BMO - 1979 - Round 1, Pb. 6): a(n) = 1, 10001, 100010001, 1000100010001,...; there are no prime numbers in this infinite sequence: A330135
- British (BMO - 1991 - Round 1, Pb. 1): a(n) = 3^n + 2 * 17^n for n >= 0 is never a perfect square : A333385
- British (BMO - 1992 - Round 1, Pb. 1): nonnegative integers k such that k and k^2 have the same number of nonzero digits: A328780
- British (BMO - 1992 - Round 1, Pb. 5): sequence of nonnegative integers satisfying a(n+1) > a(n) and a(a(n)) = 3*n: A003605
- British (BMO - 1994 - Round 1, Pb. 1): For any three-digit number k = hdu, f(k) = (h+d+u) + (h*d+d*u+u*h) + (h*d*u). This sequence consists of the numbers k for which the ratio k/f(k) is an integer: A328864
- British (BMO - 2011/2012 - Round 1, Pb. 2): a(n) is the largest integer t such that the numbers 1, 2, ..., n can be arranged in a row so that all consecutive terms differ by at least t: A004526
- British (BMO - 2016/2017 - Round 1, Pb. 6): smallest positive integer m such that m, m+1, m+2, m+3 are divisible by 2n+1, 2n+3, 2n+5, 2n+7 respectively: A279259
- France (2005 - Sélections pour IMO 2006 - Exercice 1): when prime(n) is an odd prime (n >= 2) and N(n) / D(n) = Sum_{k=1..prime(n)-1} 1/k^3, then prime(n) divides N(n) and a(n) = N(n) / prime(n): A330014
- France (2007 - Concours général - Exercice 3): Integer-sided triangles with two perpendicular medians: A335034, A335035 A335036, A335347, A335348, A335273, A335418
- Iberoamerican (1994 - Pb. 1): número natural "sensato" = Brazilian numbers: A125134
- IMO (1992 - Moscow - Pb. 6): a(n) is the greatest integer such that, for every positive integer k <= a(n), n^2 can be written as the sum of k positive square integers: A309778
- IMO (2004 - Athens - Pb. 6): multiples of 20 are exactly those integers which do not have a multiple whose decimal digits are of alternating parity: A008602
- IMO (2004 - Athens - Pb. 6): alternators = positive integers that have a multiple whose the parity of its digits alternates in base-10: A110303
- IMO (2005 - Mérida - Pb. 4): a(n) = 2^n + 3^n + 6^n - 1; 1 is the only positive integer that is relatively prime to every term of the sequence: A330170
- Italy (Gara nazionale - 1999 - Ex. 2): squarefree numbers with as many decimal digits as distinct prime factors: A167050
- Japan (1993 - Pb. 2): a(n) = 2^n – 2; these terms are the solutions of the equation 3 * A135013(x) = 2 * A000217(x): A000918
- Moscow (2001 - Level B - Pb. 5): Positive integers that are equal to 99...99 (repdigit with n digits 9) times the sum of their digits: A328683
- Moscow (2003 - Level A - Pb. 2): numbers without zero digits such that after adding the product of its digits to it, a number with the same product of digits is obtained: A327750
- Moscow (2004 - Level D - Pb. 3): a(n) is the smallest positive integer divisible by n such that it is possible to strike out a certain digit d (not a trailing zero) from its decimal expansion so that the number thus obtained will also be divisible by n and nonzero: A309631
- Moscow (2004 - Level D - Pb. 3): a(n) is the smallest positive multiple of n whose decimal expansion includes a digit (other than a trailing zero) whose removal yields a proper multiple of n: A332876
- Peru (1998 - Pb. 2) : numbers which can be expressed as sum of distinct triangular numbers: A061208
- West Germany (1981 - 1st round - Pb. 4): a(n) = 2^prime(n) + 3^prime(n) is never a perfect power: A135172
- West Germany (1982 - 2nd round - Pb. 4): a(n) = 1^n+2^n+4^n; let n>1, if a(n) is a prime number then n is the form 3^h: A001576

omega(n), number of distinct primes dividing n: A001221

Omega(n), total number of primes dividing n: A001222

one local maximum, arrays with: A007846, A000079, A087518, A087783*, A087923-A087932

one odd, two even, etc.: A001614

One potato, two potato, ...: see Josephus Problem

one puddle: see one local maximum

ones-counting sequence: A000120

open problems: see also unsolved problems in number theory (selected)

open problems: try searching in the OEIS for the following words: conjecture, apparently, appears, seems, probably, etc.

operational recurrences: A001577*

Opmanis's nice base-dependent sequence: A177834

optimal rulers: see perfect rulers

OR(x,y): A003986*

OR: A007460, A006583

orchard problem: A003035*, A006065, A008997

** order or orders, sequences related to : **

- order, binary: A029837
- order, multiplicative order of 2 mod n: A002326
- order, ord(x,y): the multiplicative order of x mod y, see entries under: multiplicative order
- order: see also under multiplicative order
- ordered factorizations: A074206*, A002033
- ordered partitions: see also under partitions
- orders, total: see total orders
- orders, weak: A000790
- orders: A000670, A004123, A004122, A004121
- orders: see also hierarchies

ordinals: A005348

Ore numbers: A001599*, A001600

Origami: A002580, A003239, A005109, A023896, A066840, A078099, A115342, A115618, A116967, A152549, A156209, A212596, A244951, A282600, A282601, A304960

** orthogonal arrays, sequences related to : **

- orthogonal arrays, number of: A039931*, A039927*, A048885*
- orthogonal arrays, see also: A008286, A039930, A048164, A048638, A048893, A049082, A049083

orthogonal groups: A003053*, A001051

out-points: A003025, A003026

overpartitions: A015128

- This is a section of the Index to the
**OEIS®**. - For further information see the main
**Index to OEIS**page. - Please read Index: Instructions For Updating Index to OEIS before making changes to this page.
- If you did not find what you were looking for in this Index, you can always search the database for a particular word or phrase.
- Full list of sections:

[ Aa | Ab | Al | Am | Ap | Ar | Ba | Be | Bi | Bl | Bo | Br | Ca | Ce | Ch | Cl | Coa | Coi | Com | Con | Cor | Cu | Cy | Da | De | Di | Do | Ea | Ed | El | Eu | Fa | Fe | Fi | Fo | Fu | Ga | Ge | Go | Gra | Gre | Ha | He | Ho | Ia | In | J | K | La | Lc | Li | Lo | Lu | M | Mag | Map | Mat | Me | Mo | Mu | N | Na | Ne | Ni | No | Nu | O | Pac | Par | Pas | Pea | Per | Ph | Poi | Pol | Pos | Pow | Pra | Pri | Pro | Ps | Qua | Que | Ra | Rea | Rel | Res | Ro | Ru | Sa | Se | Si | Sk | So | Sp | Sq | St | Su | Sw | Ta | Te | Th | To | Tra | Tri | Tu | U | V | Wa | We | Wi | X | Y | Z | 1 | 2 | 3 | 4 ]