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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). A cube has all multiplicities divisible by 3 (it is of the form a 3 for some a). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 ...
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 odd friend may be less than the even one, as in 84729645 and 155315394 (abundancy of 896/351), or in 6517665, 14705145 and 2746713837618 (abundancy of 64/27). A square number can be friendly, for instance both 693479556 (the square of 26334) and 8640 have abundancy 127/36 (this example is credited to Dean Hickerson).
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.
All pairs of positive coprime numbers (m, n) (with m > n) can be arranged in two disjoint complete ternary trees, one tree starting from (2, 1) (for even–odd and odd–even pairs), [10] and the other tree starting from (3, 1) (for odd–odd pairs). [11] The children of each vertex (m, n) are generated as follows:
Although all amicable pairs up to 10,000 are even pairs, the proportion of odd amicable pairs increases steadily towards higher numbers, and presumably there are more of them than of the even amicable pairs (A360054 in OEIS). Gaussian integer amicable pairs exist, [14] [15] e.g. s(8008+3960i) = 4232-8280i and s(4232-8280i) = 8008+3960i. [16]
Micro Mini Highland Cows are even smaller than 36 inches high. A midsize Mini can reach 42 to 48 inches high. Although on average most of these animals are between 36 and 42 inches tall ...
More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely many odd deficient numbers. There are also an infinite number of even deficient numbers as all powers of two have the sum ( 1 + 2 + 4 + 8 + ... + 2 x -1 = 2 x - 1 ).