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In number theory, a narcissistic number [1] [2] (also known as a pluperfect digital invariant (PPDI), [3] an Armstrong number [4] (after Michael F. Armstrong) [5] or a plus perfect number) [6] in a given number base is a number that is the sum of its own digits each raised to the power of the number of digits.
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4]
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
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For example 111111111111111 (15 digits) is divisible by 111 and 11111 in that base. If a number m can be expressed as a string of prime length to some base, such a number may or may not be prime, but commonly is not; for example, to base 10, there are only three such numbers of length less than 100 (1 is by definition, not prime). The three are:
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1001 1000 value1, value2 → result compare two doubles, 1 on NaN dcmpl 97 1001 0111 value1, value2 → result compare two doubles, -1 on NaN dconst_0 0e 0000 1110 → 0.0 push the constant 0.0 (a double) onto the stack dconst_1 0f 0000 1111 → 1.0 push the constant 1.0 (a double) onto the stack ddiv 6f 0110 1111 value1, value2 → result
A number that is not part of any friendly pair is called solitary. The abundancy index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a friendly number if there exists m ≠ n such that σ(m) / m = σ(n) / n. Abundancy is not the same as abundance, which is defined as σ(n) − 2n.