Expression syntax
The search box takes far more than a plain number. You can type an arithmetic
expression and the server evaluates it to a single integer, then looks that number up
(or, with Create, adds it). Whitespace is ignored, and multiplication may be left
implicit: 2(3+4) means 2*(3+4). Reading never stores anything.
Every result is capped at about 10 000 000 digits, and each function below has a fixed input ceiling chosen so a single evaluation stays near 3 seconds. Over the ceiling the request is declined rather than run.
Operators
| Operator | Meaning | Example |
|---|---|---|
| + - | Addition, subtraction | 2^100-2^50+1 |
| * / | Multiplication, exact division (the result must be a whole number) | (2^100+1)/3 |
| ^ | Power (right-associative: 2^3^2 = 2^(3^2)) | 2^127-1 |
| % | Modulo (remainder) | 2^100%1000000007 |
| \ | Integer division: a\b = ⌊a/b⌋, the quotient rounded down (same precedence as * / %) | 2^100\7+1 |
| ( ) | Grouping | (10^500-1)/9 |
Intermediate values may be negative, but the final result may not - 3-5 is
rejected. There is no unary minus; write a negative argument as 0-1 where one is
needed (see the functions below).
Shortcuts
| Form | Meaning | Example |
|---|---|---|
| Mn | Mersenne number, 2n-1 | M127 |
| Fn | Fermat number, 22n+1 | F5 |
| n! | Factorial | 1000! |
| n# | Primorial - product of all primes ≤ n | 100# |
| n## n~ | Product of the first n primes | 10## |
| I(n) fib(n) | Fibonacci number | I(5000) |
| Ln lucas(n) | Lucas number | lucas(5000) |
| b,n+ b,n- | Shorthand for bn±1 | 2,127- |
| #id | The stored number with that database id (used inside a larger expression) | #1100000000000000001/17 |
Named functions
Each is written NAME(argument, …). Every argument may itself be an expression,
so Q(5+5) and 2^Q(50)-1 are fine, and functions nest:
Q(Z(10)). The right-hand column is the largest input accepted (matched to the
~3 second budget); for the families marked † a very large parameter is also
declined early when the result would exceed the digit limit.
| Function | Meaning | OEIS | Example | Max input |
|---|---|---|---|---|
| Q(n) | Perrin number | A001608 | Q(1000) | n ≤ 30 M |
| Z(n) | Motzkin number | A001006 | Z(500) | n ≤ 700 k |
| &(n) | Narayana's cows | A000930 | &(1000) | n ≤ 30 M |
| $(n) | Padovan number | A000931 | $(1000) | n ≤ 40 M |
| K(n) | Sum of factorials, 1!+2!+…+n! | A007489 | K(100) | n ≤ 2 M |
| A(n) | Alternating factorial, n!-(n-1)!+… | A005165 | A(100) | n ≤ 2 M |
| N(m,n) | Multifactorial n·(n-m)·(n-2m)·… (m=1 is the factorial, m=2 the double factorial) | - | N(2,7) | n ≤ 4 M |
| H(m,n) | Metallic m-Fibonacci: F=m·F-1+F-2 (m=1 Fibonacci, m=2 Pell) | A000045/A000129 | H(2,100) | † n ≤ 50 M |
| Y(m,n) | m-step Lucas (sum of the previous m terms) | A001644… | Y(3,100) | m3·n ≤ 5.4·108 |
| U(n,p,q) | Lucas sequence Un(p,q): U=p·U-1-q·U-2, U0=0, U1=1 | - | U(20,1,0-1) | † n ≤ 50 M |
| V(n,p,q) | Lucas sequence Vn(p,q): V=p·V-1-q·V-2, V0=2, V1=p | - | V(20,1,0-1) | † n ≤ 50 M |
| R(m,n) | Repunit - n ones in base m, i.e. (mn-1)/(m-1) | A002275 | R(10,5) | † n ≤ 20 M |
| S(m,n) | Smarandache - concatenate 1,2,…,n written in base m | A007908 | S(10,6) | n ≤ 2.5 M |
| @(n) | The n-th prime (1-based: @(1) is 2) | A000040 | @(1000000) | n ≤ 6.5·1012 |
The Lucas U/V parameters p and q may be negative; since there is no unary
minus, write them as a subtraction - U(20,1,0-1) is the Fibonacci numbers
(p=1, q=-1) and V(20,1,0-1) the Lucas numbers.
Repunit and Smarandache bases must be at least 2, and the multifactorial step at least 1.