Category
Sequences
Primes, Fibonacci, perfect numbers and the rest of the classical integer sequences.
44 generators
Counting numbers
One, two, three and onward — the natural numbers in order, from any starting point, at any stride.
A000027Even numbers
The multiples of two, from any starting point and at any even stride — the oldest classification a number can have.
A005843Odd numbers
The integers two will not divide, from any starting point — the gnomons that build the squares, and the half of parity that folklore kept.
A005408Prime numbers
The numbers with no divisors but themselves and one — listed in order from anywhere you like, by segmented sieve.
A000040Fibonacci numbers
Each term is the sum of the two before it — the sequence behind spirals, sunflowers and the golden ratio.
A000045Square numbers
The perfect squares 1, 4, 9, 16 — the running totals of the odd numbers, and the oldest figurate sequence of all.
A000290Cube numbers
The perfect cubes 1, 8, 27, 64 — whose running totals are, remarkably, always perfect squares.
A000578Triangular numbers
The running totals of 1, 2, 3, 4 … — the counts that stack into a filled triangle, from bowling pins to handshakes.
A000217Powers of two
1, 2, 4, 8, 16 — the sequence computing is built on. In decimal, binary or hex, exact however far you take it.
A000079Factorials
n! — the product of every whole number up to n, and the count of ways to put n things in order. Exact at any size.
A000142Digits of pi
The decimal expansion of π, computed live to as many places as you like — not copied from a lookup table.
A000796Unit fractions
1, 1/2, 1/3, 1/4 — the reciprocals of the counting numbers, as fractions or as exact decimals, with the harmonic total.
Halving sequence
1, 1/2, 1/4, 1/8 — repeated halving as fractions or exact decimals, with the running total that creeps up on 2 and never arrives.
Composite numbers
Everything that is not prime and not 1 — optionally with the prime factorisation that makes it composite.
A002808Perfect numbers
Integers that equal the sum of their own divisors — 6, 28, 496, 8128. Only 52 are known, and nobody knows whether an odd one exists.
A000396Collatz sequence
Halve it when it is even, triple it and add one when it is odd — then watch the hailstone numbers climb and crash on their way down to 1.
A006577Twin primes
Primes that come two apart — (3, 5), (11, 13), (17, 19) — and the open conjecture that they never run out.
A001359Digits of the golden ratio
The decimal expansion of φ = (1 + √5)/2, computed on request from an exact integer square root.
A001622Digits of e
The decimal expansion of Euler's number, summed from 1/k! on request to as many places as you want.
A001113Lucas numbers
Fibonacci’s rule from a different start: 2, 1, 3, 4, 7, 11 — the sequence Édouard Lucas used to hunt for primes.
A000032Catalan numbers
One sequence, dozens of meanings: balanced brackets, binary tree shapes, ways to cut a polygon into triangles.
A000108Mersenne primes
Primes one less than a power of two. Only 52 are known, and the largest has over 41 million digits.
A000668Happy numbers
Square the digits, add them up, repeat. Reach 1 and the number is happy; otherwise you fall into an eight-number loop.
A007770Armstrong numbers
Numbers equal to the sum of their own digits raised to the power of how many digits they have. Only 88 exist in base 10.
A005188Palindromic primes
Primes that read the same backwards — 2, 3, 5, 7, 11, 101, 131, 151 and on. Only 11 has an even number of digits.
A002385Prime gaps
The distances between consecutive primes: 1, 2, 2, 4, 2, 4, 2, 4, 6 — including the record-setting gaps.
A001223Divisor counts
How many divisors each number has — the function whose average value Dirichlet pinned down in 1849 and whose error term is still open.
A000005Divisor sums
Add up a number’s divisors and you get σ(n) — the function that defines perfect, abundant and amicable numbers.
A000203Euler totient values
φ(n) counts the numbers below n that share no factor with it — the function at the heart of Euler’s theorem and RSA.
A000010Abundant numbers
Numbers whose divisors add up to more than the number itself — 12, 18, 20, 24. Roughly one integer in four is abundant.
A005101Deficient numbers
Numbers whose divisors add up to less than the number itself — every prime, every prime power, and about three integers in four.
A005100Amicable pairs
Two numbers that each add up to the other: the divisors of 220 sum to 284, and the divisors of 284 sum to 220.
Highly composite numbers
Numbers with more divisors than every smaller number — 1, 2, 4, 6, 12, 24, 36, 48, 60, 120 and on upwards.
A002182Pentagonal numbers
1, 5, 12, 22, 35 — the figurate numbers whose generalised form controls how every integer can be partitioned.
A000326Hexagonal numbers
1, 6, 15, 28, 45 — every one of them also a triangular number, and the centred form is the shape of a honeycomb.
A000384Tribonacci numbers
Fibonacci with a longer memory: every term sums the three before it, giving 0, 1, 1, 2, 4, 7, 13, 24.
A000073Pell numbers
Double the last term and add the one before: 0, 1, 2, 5, 12, 29, 70 — the sequence that approximates √2.
A000129Digits of the square root of 2
The decimal expansion of √2 — the first number proved irrational — from an exact integer square root.
A002193Fermat numbers
F(n) = 2^(2^n) + 1. Fermat thought they were all prime; Euler found a factor of the sixth one and ended the idea.
A000215Sophie Germain primes
Primes p where 2p + 1 is prime too. Germain invented them to attack Fermat's Last Theorem; cryptography now runs on them.
A005384Bell numbers
How many ways can you split a set into groups? 1, 1, 2, 5, 15, 52, 203 — the counts of set partitions.
A000110Kaprekar numbers
Square the number, cut the square in two, add the halves back together and get the number you started with — 45² = 2025 and 20 + 25 = 45.
A006886Automorphic numbers
Numbers that reappear at the end of their own square: 5² = 25, 76² = 5776, 9376² = 87,909,376. They go on forever, one digit at a time.
A003226Vampire numbers
Numbers that split into two equal-length factors built from their own digits — 1260 = 21 × 60. Clifford Pickover named them in 1994.
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