Über Teilerrechner: alle Teiler einer Zahl
A factor of n is a whole number that divides n with no remainder, and factors come in pairs d × (n ÷ d). The calculator finds the prime factorization first, by trial division and Pollard's rho method, and builds every factor from it: a factor of 48 = 2⁴ × 3 uses 0 to 4 twos and 0 or 1 three, which gives (4 + 1)(1 + 1) = 10 factors.
The same factorization gives the sum of the factors. Subtracting n leaves the sum of the proper factors, which decides the class that goes back to Euclid and Nicomachus: perfect when it equals n (28 = 1 + 2 + 4 + 7 + 14), abundant when it is larger (48 has 76) and deficient when it is smaller, as every prime is.
Numbers up to 10¹⁸ are accepted and factored in a fraction of a second. Up to 5,000 factors are listed in full; beyond that only their count and sum are given. For a negative number the factors come in ± pairs.
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How do you find all the factors of a number?
Test each whole number from 1 up to the square root; every divisor d you find brings its partner n ÷ d. For 48 the square root is about 6.9, and 1, 2, 3, 4 and 6 divide it, giving the pairs 1 × 48, 2 × 24, 3 × 16, 4 × 12 and 6 × 8. That is 10 factors: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. For large numbers, factor into primes first and combine them.
How do you count the factors of a number?
Write the prime factorization, add 1 to each exponent and multiply. 48 = 2⁴ × 3¹ has (4 + 1)(1 + 1) = 10 factors, and 10¹⁵ = 2¹⁵ × 5¹⁵ has 16 × 16 = 256. The count is odd only for perfect squares, because their square root pairs with itself: 36 has 9 factors, with 6 × 6 in the middle.
What is a perfect number?
A number equal to the sum of its proper factors, the factors other than itself. 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are the first two; the next are 496, 8,128 and 33,550,336. Euclid showed that 2ᵖ⁻¹(2ᵖ − 1) is perfect whenever 2ᵖ − 1 is prime, and Euler proved every even perfect number has that form. Whether an odd perfect number exists is still unknown.
What are abundant and deficient numbers?
A number is abundant when its proper factors add up to more than it, and deficient when they add up to less. 12 is the smallest abundant number, since 1 + 2 + 3 + 4 + 6 = 16. Every prime is deficient, because its only proper factor is 1. Most small numbers are deficient; the smallest odd abundant number is 945, whose proper factors sum to 975.
What is the difference between factors and prime factors?
Factors are all the numbers that divide n; prime factors are the primes among them, and multiplying them with their repeats gives n. 48 has 10 factors but only two prime factors, 2 and 3, and its prime factorization is 2⁴ × 3. The prime factorization is unique, by the fundamental theorem of arithmetic, and every factor is a product of some of those primes.
Wie genau arbeitet „Teilerrechner: alle Teiler einer Zahl“?
Die Genauigkeit hängt von Ihren Eingaben und den Annahmen der Methode ab. Die Dezimalrechnung nutzt 50 signifikante Stellen, doch Schätzungen, numerische Verfahren und Quelldaten können ungenauer sein. Die angezeigte Rundung beseitigt diese Grenzen nicht. Anhand unabhängiger Quellen geprüfte Rechenbeispiele: 10. Beispielsweise wird „48 (default, Calculator Soup example)“ anhand von Calculator Soup factors calculator: the 10 factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; σ(48) = 124 by Python divisor enumeration geprüft.
Woher stammt die Methode?
Hardy & Wright, An Introduction to the Theory of Numbers, §16.7 (the divisor functions d(n) and σ(n)) and §16.8 (perfect numbers); OEIS A000396: Perfect numbers; OEIS A005231: Odd abundant numbers; Wolfram MathWorld: Abundant number.