bibkey: sycamore2025a384309
authors: David James Sycamore
year: 2025
title: A384309 — the leading decimal digit counter sequence
doi: null
url: https://oeis.org/A384309
claim: The entry conjectures that every positive integer occurs nine times, except 1 which occurs ten times.
strata_touched:
  - D5/S3/Arith/CounterSequences/LeadingCounter
license: citation-only
triage: anchor

A384309

Verified locator

url: https://oeis.org/A384309

doi: null

Revision 32 of https://oeis.org/A384309/internal is dated 2025-07-21 and credits David James Sycamore, 2025-05-25. The exact multiplicity is labelled a conjecture. Its examples and Python generator increment the current term’s counter before reading the next term. The entry supplies no proof and cites no research paper for this claim.

Statement and proof

Start at a(1)=1. For a positive position t, count positions 1 through t whose terms have the same leading decimal digit as a(t). That count is a(t+1). The theorem proves that each positive k has a finite occurrence set, of size nine plus one when k=1.

There are nine digit classes. At least one occurs infinitely often by the infinite pigeonhole principle. Its successive visits emit every positive integer as a counter value. Each digit class contains infinitely many positive integers, witnessed by d times powers of ten for 1<=d<=9. Consequently every class is visited infinitely often. For each positive k, each class emits k exactly once, since its counter increases by one at each visit. These are nine distinct successor positions. The initial 1 adds the tenth occurrence of 1.

Proof scope

The proof reuses standard digit, pigeonhole, count, nth, and set-cardinality theorems. A000030 defines initial digits. A248034 counts occurrences of all digits and selects the last-digit counter, a different process. The separate last-k-digit-position conjecture is not proved. No publication priority or exhaustive literature coverage is claimed.