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d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
The result must be divisible by 11. 627: 62 − 7 = 55 = 5 × 11. Add 10 times the last digit to the rest. The result must be divisible by 11. (Works because 99 is divisible by 11). 627: 62 + 70 = 132: 13 + 20 = 33 = 3 × 11. If the number of digits is even, add the first and subtract the last digit from the rest. The result must be divisible ...
That is, although 360 and 2520 both have more divisors than any number twice themselves, 2520 is the lowest number divisible by both 1 to 9 and 1 to 10, whereas 360 is not the lowest number divisible by 1 to 6 (which 60 is) and is not divisible by 1 to 7 (which 420 is).
the sequence of exponents must be non-increasing, that is ; otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors (for instance 18 = 2 1 × 3 2 may be replaced with 12 = 2 2 × 3 1; both have six divisors).
There is a leap year in every year whose number is divisible by 4, but not if the year number is divisible by 100, unless it is also divisible by 400. So although the year 2000 was a leap year, the years 1700, 1800, and 1900 were common years.
The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither ...
Graph of number of -digit polydivisible numbers in base 10 () vs estimate of (). Let be the number of digits. The function () determines the number of polydivisible numbers that has digits in base , and the function () is the total number of polydivisible numbers in base .
12 (twelve) is the natural number following 11 and preceding 13. Twelve is the 3rd superior highly composite number , [ 1 ] the 3rd colossally abundant number , [ 2 ] the 5th highly composite number , and is divisible by the numbers from 1 to 4 , and 6 , a large number of divisors comparatively.