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In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. [1] [2] For example, a 7-smooth number is a number in which every prime factor is at most 7. Therefore, 49 = 7 2 and 15750 = 2 × 3 2 × 5 3 × 7 are both 7-smooth, while 11 and 702 = 2 × 3 3 × 13 are not 7-smooth.
The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 3 3 × 5 2 × 7 2 × 11 × 13 × 17 × 19 × 23 × 29. [8] If p = (p 1, ..., p n) is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant. A necessary and sufficient condition for this is that the product of p i /(p i − ...
For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned. 1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.
An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 (sequence A005408 in the OEIS). All integers are either even or odd. A square has even multiplicity for all prime factors (it is of the form a 2 for some a). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS).
Let Δ be a negative integer with Δ = −dn, where d is a multiplier and Δ is the negative discriminant of some quadratic form. Take the t first primes p 1 = 2, p 2 = 3, p 3 = 5, ..., p t, for some t ∈ N. Let f q be a random prime form of G Δ with ( Δ / q ) = 1. Find a generating set X of G Δ.
A negative base (or negative radix) may be used to construct a non-standard positional numeral system.Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base b is equal to −r for some natural number r (r ≥ 2).
In number theory, the prime omega functions and () count the number of prime factors of a natural number . Thereby (little omega) counts each distinct prime factor, whereas the related function () (big omega) counts the total number of prime factors of , honoring their multiplicity (see arithmetic function).
Squares are always congruent to 0, 1, 4, 5, 9, 16 modulo 20. The values repeat with each increase of a by 10. In this example, N is 17 mod 20, so subtracting 17 mod 20 (or adding 3), produces 3, 4, 7, 8, 12, and 19 modulo 20 for these values. It is apparent that only the 4 from this list can be a square.