This site is supported by donations to The OEIS Foundation.

# 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
- Austria (1987 – Final round – Pb. 4): Number of solutions for x+y = n and 2*x*y = z^2 : A339377
- Benelux (2011 - Luxembourg - Pb. 1): 'Benelux pairs': pairs of integers (k, m) with 1 < k < m such that k has the same prime divisors as m, and, k+1 has the same prime divisors as m+1 : A343101
- British (BMO - 1976 - Round 1, Pb. 4): a(n) = 19 * 8^n + 17 for n >= 0 is never a prime number: A330770
- British (BMO - 1977 - Round 1, Pb. 6): a(n) is the number of possible decompositions of the polynomial n * (x + x^2 + … + x^q), where q>1, into a sum of k polynomials, not necessarily all different; each of these polynomials is to be of the form b_1 * x + b_2 * x^2 + ... + b_q * x^q where each b_i is one of the numbers 1, 2, 3, ..., q and no two b_i are equal : A337566
- British (BMO - 1978 - Round 1, Pb. 3): a(1) = 1, a(1) < a(2) and a(n)^3 + 1 = a(n-1) * a(n+1), for n > 1 : A003818
- 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 - 1984 - Round 1, Pb. 4): Number of solutions of x^2 - [x^2] = (x - [x])^2 in the interval [1, n], then [0, n] : A002061, A014206
- British (BMO - 1985 - Round 1, Pb. 6): a(n) is the number of solutions of the Diophantine equation x^2 + y^2 = z^5 + z, gcd(x, y, z) = 1, x <= y : A340129, A008784
- 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 – 1993 - Round 1, Pb. 1): Squares that are concatenation of two consecutive nonzero numbers: A030466
- British (BMO - 1993 - Round 1, Pb. 2): length of the shortest straight cut which divides a right isosceles triangle (1, 1, sqrt(2)) into two parts of equal area : A154747
- 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 - 1996 - Round 1, Pb. 4): Positive integers n for which a(n) > a(n+1) when a(n) = floor(n/floor(sqrt(n))) : A079643
- British (BMO - 2007/2008 - Round 1, Pb. 2): a(n-1) is the number of solutions in positive integers (x, y, z) to the simultaneous equations (x + y – z = n, x^2 + y^2 – z^2 = n) for n>1: A063440
- 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
- British (BMO - 2020 - Round 2, Pb. 3): A131130
- Canada (1972 - Pb. 10): a(n) is the largest possible number of n-digit integers that can be in geometric progression with common ratio > 1: A341051, A341052, A341053
- China (2010 - Changhua, Taiwan - 1st day, Pb. 1): Three-digit numbers abc such that the quadratic equation ax^2 + bx + c = 0 has a rational root: A348139
- Finnish (High School Mathematics Contest - 1997 - Final round, Pb. 4) : a(n) = sum of all the n-digit numbers whose digits are all odd: A192107
- France (1990 - Concours général - Exercice 3): positive integers k such that there exist k integers x_1, x_2, ..., x_k, distinct or not, satisfying 1 = 1/(x_1)^2 + 1/(x_2)^2 + ... + 1/(x_k)^2: A074764
- 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
- France (2012 - Concours Général - Problème 1): If n = Product (p_j^k_j) then a(n) = Product (k_j^p_j): A008477
- Hungary = Eötvös-Kürschák competition (1985, Class 9-12, Category 1, Round 1, Pb. 2): a(n) = power-sum of n: A339378
- Iberoamerican (1994 - Pb. 1): número natural "sensato" = Brazilian numbers: A125134
- IMO (1990 - Beijing - Pb. 2 submitted by USSR but not used in the competition) : a(n) is the number of n-digit primes with digital product = 7: A107693, A346274
- IMO (1991 - Sweden - Pb. 2) : numbers n such that the phi(n) numbers in [1,n-1] and coprime to n form an arithmetic progression: A067133
- 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 (1994 - Hong Kong - Pb. 3): a(n) is the number of integers in the set {n+1,n+2, . . . ,2n} whose representation in base 2 contains exactly three digits 1’s: A340068, A340161
- IMO (1998 – Taipei – Pb. 3): tau(k^2)/tau(k) = integer m: A217584, A339055, A339056
- IMO (2001 - Hong Kong, Preliminary Selection Contest - Pb. 2): a(n) = number of ways n! can be expressed as the product of two coprime integers p and q such that 0 < p/q < 1: A048656
- 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
- IMO (2015 - Thailand - Pb. 2): triples (x, y, z) of positive integers for which xy – z, yz – x, and zx – y are powers of 2: A280945
- Irish (1996 - Pb. 1): a(n+1) = gcd((n+1)!, n!+1): A089026
- 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
- Kazakhstan (National Olympiad - 2015 - Day 1, problem 2): Solutions of the Diophantine equation x^y * y^x = (x+y)^z with 1 <= x <= y: A058891, A348332
- Middle European Mathematical Olympiad 2010: A001638
- Moscow (Mathematical Festival - 2008 - 6th grade, Pb. 4): Sides s of squares that can be tiled with squares of two different sizes so that the number of large or small squares is the same = A344330, A344331, A344332, A344333, A344334, A345285, A345286, A45287
- Moscow (Mathematical Olympiad - 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 (Mathematical Olympiad - 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, A340907, A340908
- Moscow (Mathematical Olympiad - 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 (Mathematical Olympiad - 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
- Moscow (Mathematical Olympiad - 2015 - 8th grade, Pb. 4): initial values of runs of 5 consecutive numbers all of which are squares, primes, or products of one prime and one square: A277225
- Peru (1998 - Pb. 2): numbers which can be expressed as sum of distinct triangular numbers: A061208
- Poland (2020/2021 - XVI Polish Juniors' Mathematics Olympiad - stage 1, Pb. 1): Number of pairs of n-digit squares such that the final (n-1) digits of the first square coincide with the initial (n-1) digits of the second: A344570
- Putnam Competition (1981 - 42nd - Problem B5): log(4) = Sum_{k>=1} A000120(k) / (k*(k+1)): A016627
- Slovenia (Mathematics Competition 38th - 1998 - 3rd grade, Pb. 1): if k is a natural number such that 2*k+1 and 3*k+1 are perfect squares, then k is divisible by 40: A045502
- USA (USAMO - 33rd - 2003 - Problem 1) : a(n) is the unique n-digit number with all digits odd that is divisible by 5^n: A151752
- USSR All-Union Mathematical Olympiad, 1971: A126933
- 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

##### One-term sequences

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 ]