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Polydivisible number

A base-b integer whose first i digits form a number divisible by i at every prefix length i.

Version
v1 · 2026-10-17 · History
Domain-specific #
14254
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Number Theory → Mathematics

Core Idea

A polydivisible number is a positive integer whose leading prefixes all pass a length-indexed divisibility test. Write it without leading zeroes in a fixed base b ≥ 2. For every prefix length i, the first i digits, interpreted in that base, must form a multiple of i. Decimal 102 passes: its prefixes 1, 10, and 102 divide by 1, 2, and 3 respectively.[ref-0a3b0a52ea94][ref-d1d2722457dc]

Scope of Application

The literal scope is positive positional integers in a declared base, with the following within-domain uses:

  • Decimal classification: Test every prefix; OEIS counts 20,456 positive decimal members after excluding zero.[^ref-d1d2722457dc]
  • Base-4 enumeration: Apply the identical rule with base-4 place values; 37 positive members remain after excluding zero.[^ref-2b03358d77a2]

Clarity

The final digit-count test alone is insufficient: decimal 111 divides by 3, but its prefix 11 fails division by 2. Membership depends on every earlier prefix. The defining yes-or-no Predicate belongs inside the family’s identity; an individual number is not itself a predicate.[^ref-0a3b0a52ea94]

Manages Complexity

One failed prefix rules out every extension that starts with it. An enumerator therefore extends only survivors: append digit d to a prefix valued p, calculate bp+d, and retain it when the new length divides that value. This prunes a search tree without adding a second membership rule.[^ref-2b03358d77a2]

Abstract Reasoning

For an n-digit base-b value N, the first i digits equal ⌊N/b^(n−i)⌋; membership requires i ∣ ⌊N/b^(n−i)⌋ for all 1 ≤ i ≤ n.[^ref-0a3b0a52ea94] OEIS estimates the length-k count by (b−1)b^(k−1)/k!. This heuristic does not prove that every base has a maximum length; universal finiteness remains conjectural.[ref-2b03358d77a2][ref-0a3b0a52ea94]

Knowledge Transfer

The rule transfers between radices when each prefix is reinterpreted in the new base. For example, 2220301₄ equals decimal 10801 and is polydivisible in base 4.[^ref-82932f01372a] A decimal pandigital member adds a digit-use restriction, not a new definition.[^ref-d1d2722457dc] Kaprekar Number and Arithmetic Number have different defining tests. Applying “polydivisible” to a nonnumeric staged check is only an analogy.

Relationships to Other Abstractions

Local relationship map for Polydivisible numberParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Polydivisible numberDOMAINPrime abstraction: Predicate — is part ofPredicatePRIME

Current abstraction Polydivisible number Domain-specific

Parents (1) — more general patterns this builds on

  • Polydivisible number is part of Predicate Prime

    The exact length-indexed membership predicate is an identity-bearing internal constituent of the Polydivisible number family.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Polydivisible number sits in a moderately populated region (54th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Integer Classifications & Arithmetic Functions (45 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

References

[^ref-0a3b0a52ea94]: Alec Mihailovs and contributors, “A109783: Largest length of a polydivisible number in base n,” The On-Line Encyclopedia of Integer Sequences (2005–), definition, Formula: Conjecture 1, and base-10 Example, https://oeis.org/A109783. The universal finiteness statement is explicitly conjectural. [^ref-2b03358d77a2]: Martin Renner and contributors, “A271373: Number of k-digit polydivisible numbers in base n,” The On-Line Encyclopedia of Integer Sequences (2016–), definition, Formula, base-4 Example row and Maple recurrence, https://oeis.org/A271373. The first count in each row includes zero; the displayed length-count expression is a heuristic. [^ref-82932f01372a]: Don Reble and contributors, “A109032: Largest polydivisible number in base n,” The On-Line Encyclopedia of Integer Sequences (2005–), definition and value at n=4, https://oeis.org/A109032. [^ref-d1d2722457dc]: N. J. A. Sloane and contributors, “A144688: Magic/polydivisible numbers,” The On-Line Encyclopedia of Integer Sequences (2009–), definition, Comments, Example and generators, https://oeis.org/A144688. Based on Roberto Bosch Cabrera. Its total includes zero.