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0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add" : a (0) = 0; for n > 0, a ( n ) = a ( n − 1) − n if that number is positive and not already in the sequence, otherwise a ( n ) = a ( n − 1) + n , whether or not that number is already in the sequence.
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
Second edition of the book. Neil Sloane started collecting integer sequences as a graduate student in 1964 to support his work in combinatorics. [8] [9] The database was at first stored on punched cards.
Each E series subdivides each decade magnitude into steps of 3, 6, 12, 24, 48, 96, and 192 values, termed E3, E6, and so forth to E192, with maximum errors of 40%, 20%, 10%, 5%, 2%, 1%, 0.5%, respectively. [nb 1] Also, the E192 series is used for 0.25% and 0.1% tolerance resistors. Historically, the E series is split into two major groupings:
12 (twelve) is the natural number following 11 and preceding 13.. Twelve is the 3rd superior highly composite number, [1] the 3rd colossally abundant number, [2] the 5th highly composite number, and is divisible by the numbers from 1 to 4, and 6, a large number of divisors comparatively.
An integer sequence is computable if there exists an algorithm that, given n, calculates a n, for all n > 0. The set of computable integer sequences is countable.The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.
Plot of the first 10,000 Pisano periods. In number theory, the nth Pisano period, written as π (n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats.
As of 2024, it is known that F n is composite for 5 ≤ n ≤ 32, although of these, complete factorizations of F n are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24. [5]