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The next number in the sequence (the smallest number of additive persistence 5) is 2 × 10 2×(10 22 − 1)/9 − 1 (that is, 1 followed by 2222222222222222222222 9's). For any fixed base, the sum of the digits of a number is at most proportional to its logarithm; therefore, the additive persistence is at most proportional to the iterated ...
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 ...
For example, if f is defined on the real numbers by = +, then 2 is a fixed point of f, because f(2) = 2. Not all functions have fixed points: for example, f(x) = x + 1 has no fixed points because x + 1 is never equal to x for any real number.
For a relational signature X, FO[PFP](X) is the set of formulas formed from X using first-order connectives and predicates, second-order variables as well as a partial fixed point operator used to form formulas of the form [, ], where is a second-order variable, a tuple of first-order variables, a tuple of terms and the lengths of and coincide with the arity of .
For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2 −23 or about 10 −7 in single precision, and exactly 2 −53 or about 10 −16 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.
In the string representing the integer x, the n th digit is 1 if F −n appears in the sum that represents x; that digit is 0 otherwise. For example, 24 may be represented by the string 100101001, which has the digit 1 in places 9, 6, 4, and 1, because 24 = F −1 + F −4 + F −6 + F −9 . The integer x is represented by a string of odd ...
A sum-product number in a given number base is a natural number that is equal to the product of the sum of its digits and the product of its digits. There are a finite number of sum-product numbers in any given base . In base 10, there are exactly four sum-product numbers (sequence A038369 in the OEIS): 0, 1, 135, and 144. [1]
The name of the harmonic series derives from the concept of overtones or harmonics in music: the wavelengths of the overtones of a vibrating string are ,,, etc., of the string's fundamental wavelength. [1] [2] Every term of the harmonic series after the first is the harmonic mean of the neighboring terms, so the terms form a harmonic ...