bibkey: yanev2017a051903 authors: Labos Elemer; Velin Yanev year: 2017 title: “OEIS A051903, maximum exponent in the prime factorization of n” doi: null url: https://oeis.org/A051903 claim: “%N Maximum exponent in the prime factorization of n. %F Conjecture: a(n) = a(A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - Velin Yanev, Sep 02 2017” strata_touched:

• D5/S3/Arith/YanevMaxExponentRadicalRecurrence license: citation-only triage: anchor

OEIS A051903

Labos Elemer authored the entry on December 16, 1999. The year 2017 records Velin Yanev’s conjecture relating the maximum prime exponent to A003557. For positive n, A003557(n) is n / primeRadical(n), where primeRadical = A007947 is the product of the distinct prime divisors. The frozen definition D5/S1/Deficit/AlmostAdditivity.primeRadical is reused. Division here is natural-number division.

With a n := n.primeFactors.sup n.factorization, the formal statement is

a 1 = 0 ∧ (∀ n : ℕ, 1 < n → a n = a (n / primeRadical n) + 1) ∧
  ∀ b : ℕ → ℕ, b 1 = 0 → (∀ n : ℕ, 1 < n → b n = b (n / primeRadical n) + 1) →
    ∀ n : ℕ, 0 < n → b n = a n

The convention a 0 = 0 is an out-of-range extension of the positive-index sequence. The recurrence is asserted only for 1 < n.

The third conjunct formalizes the second sentence of the %F line: the relation together with a(1) = 0 determines the sequence on positive indices. For n > 1, the quotient is positive and strictly smaller than n; strong induction shows that every natural-valued function with the same base value and recurrence agrees with a at every positive index. Its value at zero is unrestricted.

Verified locator

These verbatim locators are attributed to the orchestrator’s 2026-09-15 source check.

• URL: https://oeis.org/A051903
• %N (verbatim): Maximum exponent in the prime factorization of n.
• %F (verbatim): Conjecture: a(n) = a(A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - Velin Yanev, Sep 02 2017
• %A (verbatim): Labos Elemer, Dec 16 1999