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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
A perfect power has a common divisor m > 1 for all multiplicities (it is of the form a m for some a > 1 and m > 1). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included. A powerful number (also called squareful) has multiplicity above 1 for all prime factors. The first: 1, 4, 8, 9, 16 ...
For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(10 1000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(10 2000) ≈ 4605.2).
1706 = 1 + 4 + 16 + 64 + 256 + 1024 + 256 + 64 + 16 + 4 + 1 sum of fifth row of triangle of powers of 4 [394] 1707 = number of partitions of 30 in which the number of parts divides 30 [281] 1708 = 2 2 × 7 × 61 a number whose product of prime indices 1 × 1 × 4 × 18 is divisible by its sum of prime factors 2 + 2 + 7 + 61 [395]
In computing, a roundoff error, [1] also called rounding error, [2] is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. [3]
Visualisation of powers of 10 from one to 1 trillion. In mathematics, a power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer).
A fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that the last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented ...
The CPU time spent on finding these factors amounted to approximately 900 core-years on a 2.1 GHz Intel Xeon Gold 6130 CPU. Compared to the factorization of RSA-768, the authors estimate that better algorithms sped their calculations by a factor of 3–4 and faster computers sped their calculation by a factor of 1.25–1.67.