Index of integer sequences

On-Line Encyclopedia of Integer Sequences https://oeis.org/

List of integer sequences https://en.wikipedia.org/wiki/Category:Integer_sequences

https://en.wikipedia.org/wiki/Category:Integer_sequences

ℕⁿ⁰¹²³⁴⁵⁶⁷⁸⁹⁽⁺⁻⁼ᶧ⁾ Πᵝᵞᵟ⁽ᵋ⁾ᶿᶥᶹᵠᵡ Ωᵃᵇᶜᵈᵉᶠᵍʰⁱᶦʲᵏᶫˡᵐᶰⁿᵒᵖᵠʶʳˢᵗᶸᵘᵛʷˣʸᶻ Σ₀₁₂₃₄₅₆₇₈₉ Γᵦᵧᵨᵩᵪ Θₐᵦ𝒸𝒹ₑ𝒻𝓰 ₕᵢⱼₖ ₗ ₘ ₙ ₒ ₚ ᵩ ᵣ ₛ ₜ ᵤ ᵥ 𝓌 ₓ ᵧ 𝓏

smallcaps: ᴀʙᴄᴅᴇꜰɢʜɪᴊᴋʟᴍɴᴏᴘǫʀsᴛᴜᴠᴡxʏᴢ vars: 𝓪𝓫𝓬𝓭𝓮𝓯 𝒸 𝒹 𝒻 ∘ 𝓰 𝓌 𝓍 𝔂 𝓏 𝓤 𝓟 𝓍⁹ xⁱ R⁻¹ Mɴ Nᵢ Mₙ Nₘ Nᴍ n ∈ ℤ

Mersenne prime

Mersenne prime is a prime number that is one less than a power of two.

Mɴ = 2ⁿ-1 (where n ∈ ℤ)

If n is a composite number then so is 2ⁿ-1

Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes.

As of August 2020, 51 Mersenne primes are known - the largest known prime number, 2⁸²⁵⁸⁹⁹³³-1, is a Mersenne prime.

Perfect numbers

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.

Figurate number

https://en.wikipedia.org/wiki/Figurate_number

The term figurate number is used by different authors for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers).

The term can mean:

• polygonal number

• a number represented as a discrete r-dimensional regular geometric pattern of r-dimensional balls such as a polygonal number (for r = 2) or a polyhedral number (for r = 3).

• a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions.

The gnomon is the piece added to a figurate number to transform it to the next larger one.

Polite number

A polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. e.g. 13=6+7, 15=4+5+6

The sequence of polite numbers starts with (sequence A138591 in the OEIS): (3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33...)

The impolite numbers are exactly the powers of two.

Other positive integers are impolite.

Polite numbers have also been called staircase numbers because the Young diagrams representing graphically the partitions of a polite number into consecutive integers resemble staircases.

If all numbers in the sum are strictly greater than one, the numbers so formed are also called trapezoidal numbers because they represent patterns of points arranged in a trapezoid.

The politeness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For every x, the politeness of x equals the number of odd divisors of x that are greater than one.

Triangular number

https://en.wikipedia.org/wiki/Triangular_number

A triangular number or triangle number counts objects arranged in an equilateral triangle (thus triangular numbers are a type of figurate numbers, other examples being square numbers and cube numbers).

The n-th triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n (sort of an analog of factorial for summation).

n△ = n(n+1) / 2 = ∑ n = n₀ + n₁ + n₂ + n₃ + ... + nᵢ

The sequence of triangular numbers (sequence A000217 in the OEIS), starting at the 0th triangular number, is: (0, 1, 3, 6, 10, 15, 21, 28, 36, ...)

The difference of two positive triangular numbers is a trapezoidal number.

Factorial

n! = n (n-1)! = ∏ n = n₀ n₁ n₂ n₃ ... nᵢ