Numeral Systems & Integer Sequences

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Abstractions about how numbers are represented and generated, spanning positional and non-standard numeral systems (bijective numeration, non-integer base of numeration, decimal representation), recursively defined integer sequences (Perrin, Leonardo, Golomb numbers), and special constants or puzzles (Champernowne constant, Josephus problem, complete sequence).

22 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Amenable number — A positive integer n admitting a multiset of exactly n integers whose sum and product are both n.
• Bijective numeration — A numeral system giving every nonnegative integer exactly one finite digit string, with no zero digit or leading-zero ambiguity in its positional form.
• Champernowne constant — The real number formed by concatenating the positive integers in order in a chosen base, with the base-ten version proved normal.
• Common logarithm — The base-ten logarithm, the inverse of raising ten to a power and historically central to decimal calculation tables, scientific notation and orders of magnitude.
• Complete sequence — A sequence of natural numbers whose distinct finite subset sums represent every positive integer.
• Constant-recursive sequence — A sequence satisfying a fixed finite-order linear recurrence with constant coefficients.
• Decimal representation — A positional base-ten digit expansion of a nonnegative real number, with finite integer part and finite or infinite fractional part.
• Dudeney number — A base-dependent natural number that is a perfect cube whose digit sum equals its cube root.
• Farey sequence — The increasing sequence of reduced rational numbers between zero and one whose denominators do not exceed a chosen order.
• Geometric progression — A sequence in which every term after the first is obtained by multiplying the preceding term by one fixed common ratio.
• Golomb sequence — The nondecreasing self-describing integer sequence in which each positive integer n occurs exactly as many times as the nth term states.
• Josephus problem — The recurrence problem of locating the survivor or elimination order when positions in a circle are removed at a fixed counting interval.
• Keith number — A natural number whose base-b digits seed a k-step Fibonacci-like recurrence that later reproduces the number.
• Leonardo number — A recurrence sequence beginning with one and one in which each later term is the sum of the previous two plus one.
• Non-integer base of numeration — Represent numbers positionally with a real or complex radix that is not an integer, making admissible digits, expansion algorithms, and nonuniqueness depend on the radix.
• Panjer recursion — A recursive algorithm for computing an aggregate-loss distribution when claim counts belong to the Panjer (a,b,0) class and severities are discrete or discretized.
• Perrin number — A doubly infinite integer sequence generated from initial values 3, 0 and 2 by adding terms two and three positions earlier.
• Persistence of a number — The number of repeated applications of a specified digit operation needed for an integer to reach a fixed point, most commonly a single digit under repeated digit sums or products.
• Prouhet–Thue–Morse constant — The real number whose binary expansion is the Thue–Morse sequence.
• Recurrence relation — An equation defining terms of a sequence from earlier terms together with enough initial conditions to select a solution.
• Taxicab number — The smallest positive integer expressible as a sum of two positive cubes in a specified number of distinct unordered ways.
• Unitary method — A proportional-reasoning technique that first finds the value of one unit and then scales to the requested number of units.