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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
2.64 Supersingular primes ... is an Euler irregular pair. 149 , 241, 2946901 (OEIS ... write the prime factorization of n in base 10 and concatenate the factors ...
The same prime factor may occur more than once; this example has two copies of the prime factor When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, 5 2 {\displaystyle 5^{2}} denotes the square or second power of 5 ...
64 is the number of codons in the RNA codon table of the genetic code. 64 is the size in bits of certain data types in some computer programming languages, where a 64-bit integer can represent values up to 2 64 = 18,446,744,073,709,551,616. Base 64 is used in Base64 encoding, and other data compression formats.
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:
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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).