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➗ MathFree
Consecutive integers calculator
Find the run of consecutive integers that adds up to the total you choose: three numbers summing to 15 are 4, 5 and 6.
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➗ MathFree
B-smooth numbers list
Up to 30 there are 18 numbers whose prime factors are all 5 or less, the 5-smooth or Hamming numbers.
A
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➗ MathFree
Integer square root calculator
The integer square root of 82 is 9, because 9 squared is 81 and 10 squared is already 100. Paste any non-negative whole number, even one with dozens of digits…
F
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➗ MathFree
Mertens function calculator
Add up the Möbius function from 1 to n in one step: M(100) is 1, the exact signed balance between numbers with no repeated prime factor and an even or an odd…
K
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➗ MathFree
Perfect cube checker
Find out whether a whole number is a perfect cube and get its integer root: 27 is 3 cubed, and −8 is −2 cubed, since cubes also exist below zero.
S
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➗ MathFree
Pollard's p−1 factorization
Finds a factor of a composite number when one of its primes p has a smooth p − 1: 1003 with a bound of 16 splits into 17 and 59, since 17 − 1 is 16.
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➗ MathFree
Squarefree Number Checker
30 is squarefree because it factors as 2 × 3 × 5, each prime once, while 12 is not, since 2 squared divides it.
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➗ MathFree
Unitary divisor sum calculator
The unitary divisors of 12 are 1, 3, 4 and 12, which add up to 20, while all its divisors add up to 28.
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➗ MathFree
Convolution calculator
Slide one sequence across another and add up the products at each step: [1, 2, 3] convolved with [0.5, 0.5] gives [0.5, 1.5, 2.5, 1.5].
G
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➗ MathFree
Coordinate distance calculator
Enter two points and get the straight-line distance between them: from (1, 2) to (4, 6) it is exactly 5, because the sides of the right triangle are 3 and 4.
H
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➗ MathFree
Cototient calculator
The cototient of 36 is 24: that many numbers from 1 to 36 share a factor with 36. Enter any positive integer up to one trillion and get the exact count, n minus…
F
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➗ MathFree
Highly Composite Number Checker
12 is highly composite: it has 6 divisors, while no smaller number has more than 4. Enter any whole number up to 1,000,000 to see whether it sets a new…
B
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➗ MathFree
Jordan totient calculator
With order 1, the Jordan totient of 36 is 12, the same as Euler’s totient. Choose a higher order k and the calculator applies n^k times the product of (1 −…
P
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➗ MathFree
Perfect number checker
Check whether a number equals the sum of its proper divisors: 28 is perfect because 1, 2, 4, 7 and 14 add up to 28.
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➗ MathFree
Practical number checker
Tells you whether every whole number below n can be built from distinct divisors of n: 12 is practical, since 1, 2, 3, 4, 6 and 12 reach every value from 1 to…
S
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➗ MathFree
Sociable numbers checker
Follows the chain of proper-divisor sums from your number and tells you whether it comes back: 12496 returns to itself after 5 steps, visiting four other…
F
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➗ MathFree
Totatives list calculator
The numbers up to 12 that share no factor with 12 are 1, 5, 7 and 11, four in all. Enter a positive whole number to see the full ordered list and its count, not…
K
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➗ MathFree
Corner point calculator
Enter the objective coefficients and the corners of the feasible region and see which corner scores highest: for 3x + 5y over (0, 0), (0, 4), (3, 3) and (5, 0)…
H
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➗ MathFree
Calkin-Wilf fraction index
3/2 is the 5th fraction in the Calkin-Wilf sequence, two levels below the root 1/1. Enter a reduced fraction with numerator and denominator from 1 to one…
S
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➗ MathFree
Egyptian fraction term count
With the greedy method, 3/4 becomes 1/2 + 1/4, so it needs 2 unit fractions. Enter a proper fraction and get the number of terms plus each unit denominator in…
A
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➗ MathFree
Quadrant finder
Enter x and y and see where the point sits on the plane: (−3.5, 2) lies in quadrant II. A point with a zero coordinate is reported on its axis or at the origin…
G
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➗ MathFree
Cosine calculator
Type an angle in radians and get its cosine, rounded to twelve decimals: 1.0471975511965976, which is π/3, gives exactly 0.5.
H
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➗ MathFree
cos⁻¹(x) calculator
Enter a cosine between -1 and 1 and get the angle back: 0.5 gives 60 degrees, or 1.047197551197 radians.
G
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➗ MathFree
Bézout coefficients calculator
For 240 and 46 the greatest common divisor is 2, and 240 × (−9) + 46 × 47 gives 2, so x is −9 and y is 47.
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➗ MathFree
Two-coin representable checker
Find out if an exact total can be made from two coin values: with coins of 4 and 7, the amount 23 works as four 4s plus one 7.
K
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➗ MathFree
Mediant of Two Fractions Calculator
The mediant of 1/3 and 2/5 is 3/8: the numerators and the denominators are added separately, with no reduction, so you see exactly how the new fraction was…
B
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➗ MathFree
Negative Pell equation solver
Find out whether x² − Dy² = −1 has whole-number solutions and get the smallest one: when D is 13, the answer is x 18 and y 5, found from a continued fraction…
F
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➗ MathFree
Pierce expansion calculator
Enter a fraction between 0 and 1 and get its exact Pierce digits: 2/3 gives 1 and 3, because 1/1 − 1/(1·3) rebuilds two thirds.
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➗ MathFree
Pythagorean triples from a leg
Enter one leg of a right triangle and get every whole-number triangle that shares it: a leg of 20 has four partners, from 20-15-25 to 20-99-101.
A
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➗ MathFree
cos(2θ) calculator
Enter an angle θ in radians and get the cosine of its double, worked out as cos²θ minus sin²θ: a θ of 0.7853981634, a quarter of π, gives exactly 0.
H
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➗ MathFree
Hyperbolic cosine calculator
Get the hyperbolic cosine of any number: cosh(1.5) is 2.352409615243247. It is the average of e^x and e^-x, the curve traced by a hanging chain or power line…
G
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➗ MathFree
Distinct parts partition count
The number 5 splits into different parts in 3 ways: 5, 4+1 and 3+2. Enter any whole number from 0 to 5000 and get the exact count of its distinct-part…
A
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➗ MathFree
Gaussian Integer Division Calculator
Dividing 17 + 9i by 4 − 3i gives quotient 2 + 3i and remainder 3i, whose norm 9 is below the divisor’s 25.
K
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➗ MathFree
Gaussian integer factorization
Factor any nonzero Gaussian integer into Gaussian primes: 5 becomes −i · (1+2i) · (2+i), with the unit, exponents and norms listed.
S
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➗ MathFree
Goldbach partition count
10 splits into two prime pairs, 3 + 7 and 5 + 5, so its Goldbach partition count is 2. Enter any even number from 4 up to ten million to get that count, with…
B
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➗ MathFree
Sum of three squares representation calculator
310 can be written as 2² + 9² + 15², and the result shows those three numbers with the full equation checked.
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➗ MathFree
Sum of Two Squares Count
65 can be written as a sum of two squares in 16 ordered, signed ways, so its count is 16. Enter a non-negative whole number and get the exact total from its…
B
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➗ MathFree
Waring problem representation
With squares, 30 becomes 25 + 4 + 1, that is 5² + 2² + 1², by always taking the largest square that still fits.
P
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➗ MathFree
Cosine similarity calculator
Score how closely two vectors point the same way, from -1 for opposite to 1 for identical direction.
H
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➗ MathFree
Cullen number generator
At index 5 the Cullen number is 161, because 5 × 32 + 1 = 161. Enter any whole index from 0 upwards and get the full term with every digit exact, however long…
K
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➗ MathFree
Digit factorial chain length calculator
Starting from 69, the digit factorial chain visits 5 distinct values before one repeats; its first step is 6! + 9! = 363600.
B
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➗ MathFree
Factorion checker
145 is a factorion: 1! + 4! + 5! is 1 + 24 + 120, which adds back up to 145. Enter a positive whole number and see each digit's factorial, the total and a plain…
A
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➗ MathFree
Fermat number generator
Type an index and see the Fermat number 2^(2^n)+1 written out in full: index 0 gives 3 and index 4 gives 65537.
S
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➗ MathFree
Repunit checker
Checks whether a positive whole number is written only with the digit 1: 111 is the three-digit repunit, equal to (10^3 − 1)/9.
P
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➗ MathFree
Vampire Number Checker
1260 is a vampire number because 21 × 60 = 1260 and the digits of 21 and 60 are exactly 1, 2, 6 and 0.
F
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➗ MathFree
Additive prime checker
23 is an additive prime: it is prime, and its digits add up to 5, which is prime too. Enter a whole number from 2 and see both checks side by side, so you know…
S
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➗ MathFree
Balanced prime checker
53 is a balanced prime: the nearest primes are 47 below and 59 above, both 6 away, and their average is 53.
B
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➗ MathFree
Permutable Prime Checker
Check whether a prime stays prime however you rearrange its digits: 113 passes, since all 3 distinct orderings of its digits are prime.
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➗ MathFree
Safe prime checker
Checks whether a prime p is safe, meaning (p − 1)/2 is prime too: 23 is safe because its half, 11, is prime, which also makes 11 a Sophie Germain prime.
F
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➗ MathFree
Wieferich prime checker
1093 is a Wieferich prime: 2 raised to 1092 leaves remainder 1 when divided by 1093², which is 1194649.
P
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➗ MathFree
Dudeney number checker
512 is a Dudeney number: its digits add up to 8, and 8 cubed is 512. Enter any positive whole number to see its digit sum, whether it is a perfect cube, its…
B
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➗ MathFree
Evil number checker
3 is an evil number: in binary it is 11, with 2 ones, and 2 is even. Enter any non-negative whole number to see its binary form, how many ones it has and…
P
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➗ MathFree
Home Prime Calculator
12 factors as 2 × 2 × 3, joining those digits gives 223, and 223 is prime, so the home prime of 12 is 223.
F
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➗ MathFree
Lucas prime checker
Enter an index and see whether the Lucas number at that position is prime: L(5) is 11, a two-digit prime, while L(6) is 18 and composite.
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➗ MathFree
Mersenne prime checker using the Lucas-Lehmer test
Give a prime exponent and the Lucas–Lehmer test tells you whether 2^p − 1 is prime: for 31 it runs 29 steps, ends with a residue of 0 and confirms the prime…
K
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➗ MathFree
Prime triplet finder
Lists every prime triplet up to your limit in the two shapes p, p + 2, p + 6 and p, p + 4, p + 6: up to 50 there are 7, from (5, 7, 11) to (41, 43, 47).
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➗ MathFree
Alternating digit sum calculator
For 918082 the alternating digit sum is 22, a multiple of 11, so the number divides by 11 exactly.
F
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➗ MathFree
Balanced ternary converter
20 in balanced ternary is 1 −1 1 −1: three cubed, minus three squared, plus three, minus one. Enter any integer, negative ones included, to get its signed-digit…
B
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➗ MathFree
Digital root in base
In base 16, 255 is written FF, its digits add up to 30 and then to 15: the digital root is 15, reached in 2 steps.
S
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➗ MathFree
Equidigital Number Checker
27 is equidigital: it has 2 digits, and its factorization 3³ also uses 2, the base 3 and the exponent 3.
K
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