Niven's constant¶

The limiting average, over positive integers, of the largest exponent in each integer’s prime factorization.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5775
Origin domain: analytic number theory
Subdomain: analytic number theory

Core Idea¶

The value depends on defining H of one, averages largest rather than smallest exponent and convergence follows from the distribution of perfect-power divisibility. The largest prime exponent exceeds or equals k exactly when the integer is divisible by a nontrivial k-th power, and summing those tail probabilities produces a zeta-series for the mean. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Niven's constant belongs to analytic number theory and is useful where the analyst can specify the typed analytic number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the integer function H and H of one convention, prime factorization and maximum exponent, Cesàro average through n, limiting operation, series one plus the sum for k at least two of one minus reciprocal zeta k, convergence proof and numerical value are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the integer function H and H of one convention, prime factorization and maximum exponent, Cesàro average through n, limiting operation, series one plus the sum for k at least two of one minus reciprocal zeta k, convergence proof and numerical value are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Niven's constant. Niven's constant compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed analytic number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the integer function H and H of one convention, prime factorization and maximum exponent, Cesàro average through n, limiting operation, series one plus the sum for k at least two of one minus reciprocal zeta k, convergence proof and numerical value are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of analytic number theory because they reuse the typed analytic number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The largest prime exponent exceeds or equals k exactly when the integer is divisible by a nontrivial k-th power, and summing those tail probabilities produces a zeta-series for the mean., and type the carrier, state every parameter and convention in the definition, test that the integer function H and H of one convention, prime factorization and maximum exponent, Cesàro average through n, limiting operation, series one plus the sum for k at least two of one minus reciprocal zeta k, convergence proof and numerical value are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Niven's constant Domain-specific

Parents (1) — more general patterns this builds on

Niven's constant is a kind of Aggregation Prime

The proposed strict upward parent is prime:aggregation.

Hierarchy path (1) — routes to 1 parentless root

Niven's constant → Aggregation → Micro Macro Linkage

Neighborhood in Abstraction Space¶

Niven's constant sits in a crowded region of the domain-specific corpus (12^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Arithmetic Functions & Number Sequences (16 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08