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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 basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; [2] [3] this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that ...
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 rule is that if the year is divisible by 100 and not divisible by 400, the leap year is skipped. The year 2000 was a leap year, for example, but the years 1700, 1800, and 1900 were not.
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 ...
The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 (sequence A046760 in the OEIS). An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.
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).
In 1968 Martin Gardner noted that most even amicable pairs sumsdivisible by 9, [12] and that a rule for characterizing the exceptions (sequence A291550 in the OEIS) was obtained. [ 13 ] According to the sum of amicable pairs conjecture, as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs ...