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 numeral passes a divisibility test at every leading prefix. Fix a base b ≥ 2 and write the integer without leading zeros as d₁d₂…dₙ. For each i = 1,…,n, interpret the first i digits in base b; that prefix value must be divisible by i. The last test alone does not suffice. The rule defines a family of numbers rather than one distinguished number or one enumerating program.^[1]

For example, decimal 102 passes because its successive prefixes 1, 10, and 102 are divisible by 1, 2, and 3. The same rule also applies to base-4 numerals, with each prefix interpreted in base 4. These are changes of a parameter inside one number-theoretic family, not separate nonnumeric applications.^[2]^[3]

Structural Signature¶

Radix and canonical numeral. The carrier is a positive finite integer expressed in a fixed base b ≥ 2, with digits from 0 to b−1 and a nonzero leading digit. Changing the base can change membership; counting zero requires a separate convention.^[1]^[2]
Ordered leading prefixes. For each length i, take exactly the first i digits, not a suffix, a substring, or only the complete numeral. The prefixes are nested and every nonempty length through n is tested.^[1]
Length-indexed divisibility. The numerical value of the length-i prefix is a multiple of i. This matching of prefix length to divisor, repeated for all i, is the defining predicate. Replacing i with one fixed divisor defines a different family.^[1]

The rule permits a derived search recurrence. If a valid length-i prefix has value p, appending digit d gives bp+d; keep it only when i+1 divides bp+d. That is an exact way to enumerate members, but running this procedure is not an additional membership condition.^[4]

What It Is Not¶

A number divisible by its total digit count need not be polydivisible. Decimal 111 is divisible by 3, yet its two-digit prefix 11 is not divisible by 2. The failed earlier prefix rules it out.^[1]

Nor does the name assert that the family is finite in every base. OEIS reports finite computed trees for particular bases but explicitly labels universal finite termination a conjecture. The estimate for the count at a given length is a heuristic, not a proof that every base has a last surviving depth.^[1]^[4]

The decimal sequence record includes 0 as a one-digit “magic” value. Here the carrier is positive with a nonzero first digit, so 0 is excluded unless a separate counting convention is stated. A pandigital polydivisible numeral is a constrained subcase; using every digit once is not a required part of this identity.^[2]

Scope of Application¶

The literal scope is positional integer numerals in an integer base at least two. The defining test is the same whether one checks a short decimal example, enumerates a base-4 tree, or imposes an extra decimal digit puzzle. A statement about the number of members or maximal length must say which base and whether zero is counted.^[4]^[2]

For the documented decimal sequence, the OEIS total is 20,457 including zero, hence 20,456 positive members. The base-4 row of its count triangle sums to 38 numbers including zero, hence 37 positive members. These are computed cases with different radices and an explicit count convention; they do not establish a theorem for every radix.^[2]^[4]^[1]

Clarity¶

“Divisible by its number of digits” is ambiguous if it is heard as one final check. The complete statement has a universal quantifier: for every prefix length i from one to the full length, the first i digits, read as a base-b integer, are divisible by i. In arithmetic notation, if N has n base-b digits, the length-i prefix has value ⌊N/b^(n−i)⌋, and membership requires i ∣ ⌊N/b^(n−i)⌋ for all 1 ≤ i ≤ n.^[1]

The 111 near miss exposes the quantifier: it passes the i=3 test and fails the i=2 test. Stating the radix, leading-zero convention, and all prefix lengths makes classification reproducible rather than dependent on a visually persuasive final number.

Manages Complexity¶

The prefix test gives an exact pruning rule. Once a prefix fails at length i, every longer numeral beginning with those same digits retains that failing prefix and must fail too. An enumerator can therefore extend only surviving prefixes by the b possible next digits and test bp+d modulo i+1.^[4]

This compression is local. Every level has finitely many candidates, and a completed computation can prove a last level for a specified base if its next level is empty. Finite branching by itself does not prove that every base's tree ends at some depth; that is the separate all-base conjecture.^[1]

Abstract Reasoning¶

The rule supports a clean induction for any fixed numeral. Its first digit passes division by 1. Given a valid length-i prefix p, the next candidate has value bp+d. It is valid through length i+1 exactly when (bp+d) mod (i+1) = 0, because all shorter prefixes are unchanged. The recurrence proves the extension test, not eventual extinction of the whole tree.^[4]

One may compare depth counts for particular radices. OEIS records the heuristic T(b,k) ~ (b−1)b^(k−1)/k! for the number at length k; the factorial denominator suggests why counts might eventually fall. A heuristic approximation, however, is not an exact bound excluding rare long branches and cannot turn the stated universal conjecture into a theorem.^[4]^[1]

Knowledge Transfer¶

The exact procedure transfers among bases: reinterpret every prefix in the selected radix and retain the divisor equal to its digit length. In base 4, decimal 10801 is written 2220301₄; in base 10, 102 is a short member. Their digit alphabets and counts differ, but all three defining roles remain literal.^[3]^[2]

The method also applies to a restricted search within one base. OEIS identifies 381654729 as a decimal polydivisible numeral that uses digits 1 through 9 once each. Its pandigital restriction narrows the search; it does not redefine polydivisibility. Using “prefix gate” to discuss a nonnumerical staged process may be a helpful analogy, but without base-b numeral values and divisibility by length it is not another instance of this named family.^[2]

Examples¶

Canonical¶

Take decimal 102. The radix is 10 and the positive numeral has no leading zero. Its ordered prefixes are 1, 10, and 102; dividing by their lengths gives 1, 5, and 34. All are integers, so the number belongs to the family. If the middle prefix were changed to 11 while the final three-digit number happened to be divisible by 3, that final pass would not repair the earlier failure.^[2]

Mapped back: fixed radix and numeral → every leading prefix in order → divisibility by the matching prefix length. The number qualifies because all three roles hold at all three lengths.

Applied / In Practice¶

In base 4, 2220301₄ has decimal value 10801. Its seven prefix values in decimal are 2, 10, 42, 168, 675, 2700, and 10801. Dividing by lengths 1 through 7 yields 2, 5, 14, 42, 135, 450, and 1543, all integers. It is a complete member, and OEIS records 10801 as the largest member for this particular base.^[3]^[1]

Mapped back: base 4 fixes digit values → the seven prefixes are nested in order → each passes its corresponding divisor. This is a different radix realization of the same formal rule, not an unlike external application.

Structural Tensions¶

The inspected sources do not establish an internal trade-off with two opposed design objectives. The prefix condition is an exact predicate, and the computed end of a particular base's search tree does not compete with the conjecture that all bases terminate. They have different quantifier scopes.^[1]

The important boundary is epistemic: an extension rule and a length-count heuristic help investigate the family, but neither proves the universal termination claim. Reporting the heuristic as a theorem would obscure the very question the source leaves open.^[4]^[1]

Structural–Framed Character¶

This entry lies strongly toward the structural end of the spectrum. Its defining operation is arithmetic, and the same membership test can be applied regardless of why someone chooses a numeral.^[1]

Evaluative weight: Passing the prefix predicate is true or false by calculation; usefulness, elegance, or puzzle appeal does not decide membership.
Human practice: Mathematicians choose bases, notation and search procedures, but a numeral that passes the stated tests remains a member whether it is studied or not.
Institutional origin: A sequence database curates examples and counts, but no institution grants membership to a numeral that fails its second or any other prefix test.
Vocabulary travel: “Prefix”, “stage” and “search tree” can describe other settings. “Polydivisible number” retains the numeral and divisibility rule rather than becoming a label for any staged filter.
Import versus recognition: Checking a new base-b numeral is literal recognition. Calling an unrelated multistage screening process polydivisible would import a metaphor and leave out the defining arithmetic.

Its character: an exact structural number family whose applications may be framed as computation or puzzle solving, while its membership is fixed by prefix divisibility.

Structural Core vs. Domain Accent¶

The thin skeleton is a nested prefix rule with incremental rejection: a failed partial object bars all extensions that preserve it. Here the thicker, name-bearing relation is specific. Each prefix must be a positional integer value, and its divisor must equal that prefix's digit length. A generic staged test lacks those terms and does not instantiate a polydivisible number.^[1]

A broader cross-domain “prefix-gated family” might someday merit a separate Prime if primary sources show literal recurrence with stable roles, counterexamples and inferences in unlike nonnumeral carriers. The present evidence shows only number representations and digit puzzles. The named entry remains within arithmetic; its generic search skeleton is a future comparison question, not a claimed Prime or an excuse to remove the divisor condition.

This entry is part of Predicate.

Predicate is a truth-valued test on an object. For fixed base b, let P_b(N) be true exactly when every length-i leading prefix of N is divisible by i. This bivalent test is internal to the named number family: its true cases are precisely the polydivisible numerals in that base. Replacing it with a final-length-only test would admit decimal 111 and change the family. Predicate itself has many other extensions and uses; this family adds the particular digit and divisor rule. In the unary case, the test also has the form of a Relation on a declared numeral domain.

Classification is the act of assigning objects to categories by criteria. One can classify integers after evaluating P_b, but sorting a list is not required for a numeral to qualify. Function Mapping describes a single-valued input–output rule. The append operation p ↦ bp+d is such a mapping for fixed b,d; polydivisibility additionally requires every nested prefix to pass its indexed divisibility test. A qualifying numeral need not have been found by that particular program. Other digit-defined number families may share the integer carrier while asking different questions of its digits.

Relationships to Other Abstractions¶

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

Polydivisible number → Predicate → Relation

Neighborhood in Abstraction Space¶

Polydivisible number sits in a moderately populated region (54^th 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

Not to Be Confused With¶

The decisive near miss is a number divisible by its own digit count but not at every prior prefix: 111 is the small counterexample. A pandigital requirement checks which digits appear; it can be added to polydivisibility but does not replace the prefix tests. An algorithm that searches the tree is a method for finding members, not a member itself.^[1]^[2]^[4]

Kaprekar Number tests a square–split–sum fixed-point equation, and Arithmetic Number tests whether the mean of an integer's positive divisors is integral. Neither test is the all-prefix digit-length divisibility rule. These are neighboring catalog identities, not aliases of Polydivisible Number.

References¶

[1] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p

[2] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i

[3] 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. registry ↩a ↩b ↩c

[4] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i