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The number 4 is not untouchable, as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4. The number 5 is untouchable, as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be ...
The semiprimes are the case = of the -almost primes, numbers with exactly prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes).
The sum of one odd square and one even square is congruent to 1 mod 4, but there exist composite numbers such as 21 that are 1 mod 4 and yet cannot be represented as sums of two squares. Fermat's theorem on sums of two squares states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent ...
A route (or road) number, designation or abbreviation is an identifying numeric (or alphanumeric) designation assigned by a highway authority to a particular stretch of roadway to distinguish it from other routes and, in many cases, also to indicate its classification (e.g. motorway, primary route, regional road, etc.), general geographical location (in zonal numbering systems) and/or ...
Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.
Ordinal numbers, not cardinal numbers, are commonly used to represent dates, because they are in the format of 'in the tenth year of Caesar', etc. which also carried over into the anno Domini system and Christian dating, e.g. annō post Chrīstum nātum centēsimō for AD 100.
For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ. The only odd highly abundant numbers are 1 and 3. Relations with other sets of numbers Euler diagram of numbers under 100:
If a representation has an odd number of terms, x/y is the middle term, while if it has an even number of terms and its minimum value is m it may be extended in a unique way to a longer sequence with the same sum and an odd number of terms, by including the 2m − 1 numbers −(m − 1), −(m − 2), ..., −1, 0, 1, ..., m − 2, m − 1.
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