Metadata
bibkey: devlieger2021a347285
authors: M. De Vlieger; OEIS Foundation Inc.
year: 2021
title: OEIS A347285 descending prime-power exponents
doi: null
url: https://oeis.org/A347285
claim: For every positive row index n, the sum S of absolute first differences satisfies n - S = 1.
strata_touched:
- D5/S3/Arith/Primes/OeisA347285PrimePowerDescent
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A347285
Pinned text: https://raw.githubusercontent.com/oeis/oeisdata/e7d0073e/seq/A347/A347285.seq
OEIS A347285, revision 17, created 26 August 2021; source reading recorded in https://github.com/the-omega-institute/trureturing/issues/18689. The entry is unpaginated.
Source statements
NAME: “Irregular triangle T(n,k) starting with n followed by e_k = floor(log_p_k(p_(k-1)^e_(k-1))) such that e_k > 0.”
FORMULA: “T(n,1) = n; T(n,k) = floor(log_p_k(p_(k-1)^T(n,k-1))).”
COMMENTS: “T(0,1) = 0 by convention; 0 is not allowed for n > 0.”
COMMENTS: “Conjecture: let S be the sum of the absolute values of the first differences of terms in row n. For all n > 0, n - S = 1.”
Mathematical scope
The prime index is one-based and the exponent index in Lean is zero-based: e n j is T(n,j+1). The recurrence uses Real.logb and Nat.floor, and the positive row stops at the first zero. Integer absolute differences give a natural sum S; the conjecture subtracts that sum in the integers. The row n=0 is excluded. Strict descent, Bertrand’s postulate and telescoping prove the stated identity. The comment “The least first difference among row n is 1” and the asymptotic growth of the row length are not settled by this identity.
ASSUMED-UNVERIFIED
Literature-search completeness and the absence of a prior proof remain ASSUMED-UNVERIFIED. The source and searched collections are recorded in issue #18689; the formal theorem does not establish a priority claim.