{"id":"26eea8a4-e1e0-40fe-ac08-82a7f7a34752","arxiv_id":"2508.20853","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves a rigorous asymptotic with explicit constant for sequence A283190, bounds consecutive differences by O(log log n), and establishes lower and upper bounds for iterated remainder sets.","lead":"This paper proves that the count of distinct remainders when n is divided by integers up to n/2 grows at a linear rate with an explicit constant, and it bounds how much this count can jump between consecutive n. It also analyzes iterated remainder sets, linking them to an open problem on Pierce expansions.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"False inequality |Mp| ≤ n/(p(p+1)) in (5) undermines E1 bound; counterexample n=p^2+1.","rationale":"The reader's weakest_assumption concerned Lemma 13's base case, which is a secondary result (Theorem 3). The central claim is Theorem 1, and its proof contains a false inequality in the estimate of E1. This is more directly load-bearing for the paper's headline result. However, the flaw is readily repairable: the number of primes p with |M_p(n)| exceeding n/(p(p+1)) is at most π(√n), and their total contribution is O(√n/log n), which is negligible relative to the claimed O(n/(log n log log n)). Thus the theorem is very likely true, but the written proof is incomplete. Since the reader's verdict was already CONDITIONAL, this concern does not move it; it does, however, provide a different and more central reason for conditionality. The minor issues in Lemma 13 and Proposition 4 also support leaving the verdict unchanged.","tokens_in":12094,"tokens_out":46701,"duration_ms":384989,"concrete_test":"For a fixed prime p (e.g., p=7), take n=p^2+1=50, compute L = p(n+1)/(p+1) = 44.625, and verify p^2=49 ∈ [L,49] and 49 ∈ D_p. Then |M_p(50)| = 1 > 50/56, contradicting the inequality used in (5). To test the repair, compute S(N) = Σ_{p≥log N} |M_p(N)| for N up to 10^6 and compare with N/(log N log log N); the deviation should be O(√N/log N), confirming the final error bound survives the correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1, the error term E1 (equation (5)) is bounded using |Mp(n)| ≤ n/(p(p+1)) for p≥X. This inequality is false. For any prime p and n=p^2+1, the integer m=p^2 lies in the interval [p(n+1)/(p+1), n-1] (since L=p(p^2+2)/(p+1)<p^2), and p is the least prime factor of p^2, so p^2 ∈ Mp(n). Hence |Mp(n)| ≥ 1, while n/(p(p+1)) = (p^2+1)/(p(p+1)) = 1 - 1/(p+1) + 1/(p(p+1)) < 1. Thus the stated inequality is violated for infinitely many p. The resulting bound E1 = O(n/(X log X)) is not justified as written. The gap is repairable: the extra contribution from such p is at most the number of primes with p^2 ≤ n-1, i.e., O(√n/log n), which is absorbed by the claimed error term O(n/(log n log log n)). However, the proof as written needs this additional argument. A secondary issue is that Lemma 3 is stated for integer a,t but applied with real values; the proof works for reals, so this is only a formal blemish. These issues do not appear to invalidate the constant c or the main asymptotic, but they leave Theorem 1 not fully rigorous as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the sequence s(n), the number of distinct remainders n mod k for 1 ≤ k ≤ floor(n/2). The main result (Theorem 1) claims an asymptotic formula s(n) = c n + O(n/(log n log log n)) with c given by an explicit infinite product over primes, approximately 0.2296. The paper also proves that consecutive differences satisfy s(n+1) ≤ s(n)+1 and s(n+1) ≥ s(n) - O(log log n), and that arbitrarily large decreases occur. Finally, it introduces iterated remainder sets S_j(n) and proves upper and lower bounds of sizes n/(j+2) and n/((j+2)!) up to constants, relating them to an open problem on Pierce expansions. The proofs use elementary sieving, the prime number theorem, Dirichlet's theorem, and a recursive construction.","tokens_in":12489,"tokens_out":12632,"duration_ms":132540,"significance":"If the proofs are repaired, the paper provides the first rigorous derivation of the linear asymptotic density of s(n) with an explicit constant, confirming and quantifying a heuristic StackExchange/OEIS observation. The results on consecutive differences and iterated remainder sets are new and address a natural sequence that appears to have been little studied. A notable strength is that the constant c is defined a priori as an infinite product, with no fitted parameters; the paper also connects the iterated sets to an existing open problem and gives nontrivial bounds. The central claims are plausible and likely correct, but the current manuscript contains a few load-bearing proof gaps that must be fixed before the results can be considered fully established.","major_comments":[{"comment":"The bound |Mp(n)| ≤ n/(p(p+1)) used for E1 is false. For n = p^2 + 1, the integer m = p^2 lies in the interval [p(n+1)/(p+1), n-1] and is divisible by p but by no smaller prime, so |Mp(n)| ≥ 1, while n/(p(p+1)) < 1. Thus the displayed inequality and the resulting E1 estimate are not justified as written. The gap is repairable: for p > sqrt(n), Mp(n) is actually empty (any m = p a with a ≥ 2 has a prime factor < p, and a = 1 gives m = p < p(n+1)/(p+1)); for p ≤ sqrt(n) one may use |Mp(n)| ≤ (n-p)/(p(p+1)) + 1, and the extra sum over p ≤ sqrt(n) is O(sqrt(n)/log n), absorbed by the claimed error. This additional argument needs to be supplied.","section":"Section 3, Eq. (5)"},{"comment":"The base case is internally inconsistent. S0(n) was defined in Section 2 as {1,2,...,floor(n/2)}, which does not contain 0, yet the proof states 'For j=0 the inclusion (14) is clearly true with x0=0, since S0(n) = {0,1,...,floor(n/2)}.' The stated inclusion for j=0 requires 0 to be in S0(n). This can be fixed by starting the inclusion at r=1 for j=0 and treating the j=0 lower bound separately, or by redefining S0(n) to include 0 and adjusting the definitions. As written, the induction base for Theorem 3 is false, and this is load-bearing for the lower bound.","section":"Section 6, proof of Lemma 13"},{"comment":"The formula r' = (r-1)Q + 1 appears to be a typo; it should be r' = (r+1)Q - 1. With the printed formula, r'+1 = (r-1)Q + 2 does not in general divide n(j+1)+1 = (n(j)+1)Q, so condition (i) of Lemma 9 fails. The subsequent claim that r' = Q - 1 works is also consistent with the corrected formula. This is a load-bearing error in the construction proving liminf of differences is -infinity, though it is clearly repairable.","section":"Section 4, proof of Proposition 4"}],"minor_comments":[{"comment":"Lemma 3 is stated for positive integers a and t, but in the proof of Theorem 1 it is applied with a = n - (n-p)/(p+1) - 1 and t = (n-p)/(p+1), which need not be integers. The proof can be adapted by replacing t with floor(t) and absorbing the O(1) discrepancy into E2, but the formal mismatch should be noted.","section":"Section 3, Lemma 3 application"},{"comment":"In the proof, 'the prime factorization of n' should read 'of n+1'.","section":"Section 4, Proposition 3"},{"comment":"The caption of Figure 2 says 'plots of sj(n) for n = 1,2,3' but the text means j = 1,2,3. Please correct.","section":"Section 7, captions"},{"comment":"There are a few small textual inconsistencies, e.g., the definition of T(n) versus T(n,n+1), and the statement of Lemma 9 uses r+1 > floor((n-2)/3) in one place. These do not affect the arguments.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural and previously somewhat neglected sequence, and the main constant is explicit and parameter-free. The identified proof gaps are localized and appear repairable without changing the main results. However, the false inequality in (5), the inconsistent base case in Lemma 13, and the typo in Proposition 4 are all in load-bearing positions, so a major revision is appropriate. I have no concerns about novelty or attribution; the prior StackExchange contribution is properly acknowledged."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper proves rigorous asymptotics for s(n), the number of distinct remainders n mod k, with an explicit error term, and adds new results on consecutive differences and iterated remainder sets. The constant was already known informally from a StackExchange answer, but the proof and the error term are new, and Theorems 2 and 3 look genuinely new. The writing is clear and the tools are standard; there is no circularity or data fitting.\n\nThe main soft spot is in the proof of Theorem 1. The bound on E1 uses |M_p(n)| ≤ n/(p(p+1)), which is false. For n = p^2+1, m = p^2 lies in the interval, so |M_p| ≥ 1 while the bound is < 1; e.g., n=5, p=2. This gap is repairable: use |M_p(n)| ≤ n/(p(p+1)) + 1, and note that M_p(n) is empty for p > √n, so the +1 contributes at most π(√n) = O(√n/log n), which is absorbed by the claimed error. That is a few lines, but as written the proof is not fully rigorous.\n\nLemma 13 has a false base case: for j=0 it claims S0(n) contains {0,...,⌊n/2⌋}, but S0(n) was defined without 0. The inclusion misses only r=0, and the induction can be restarted at r=1, so this is minor. Lemma 3 is stated for integers a,t but applied to rationals; the proof works for reals, so also minor.\n\nThese are fixable issues. The difference results and iterated remainder bounds appear solid, and the paper is a good contribution to a surprisingly understudied sequence. I would send it to peer review; the authors should patch the E1 bound and the Lemma 13 base case. I'd also bring it to reading group, since the gap in the sieve bound is a nice example of why edge cases matter.","headline":"Rigorous asymptotics for a natural OEIS sequence, with two fixable proof gaps; the main results look correct and the paper deserves refereeing.","tokens_in":12907,"tokens_out":19034,"would_cite":true,"duration_ms":161882,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11A07","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that distinct remainders have an exact linear density c ≈ 0.2296, with an explicit error bound.","keywords":["number of remainders","distinct residues","asymptotic density","sieve argument","consecutive differences","iterated remainder sets","Pierce expansions","integer sequence"],"falsifier":"Verify Theorem 1 by exact enumeration: if the normalized error (s(n) − c·n)/(n/(log n log log n)) is unbounded over n up to, say, 10^8, the claimed O-term fails; the paper's own data to 10^7 already fits a much smaller error. For the iterated lower bound, the literal base case of Lemma 13 fails at n=4: the proof requires {0,1} ⊆ S0(4), but S0(4) = {1,2}.","tokens_in":12043,"feed_emoji":"🧮","tokens_out":10692,"duration_ms":111387,"temperature":0.7,"pith_summary":"Counting how many different remainders appear when a fixed n is divided by each k up to n/2 looks simple but had no rigorous asymptotic. This paper proves the count s(n) is exactly c·n plus an error of size O(n/(log n log log n)), where c ≈ 0.2296 is a concrete product over primes, confirming a density that had only been conjectured from data. It also shows that moving from n to n+1 can raise the count by at most one, can lower it by arbitrarily large amounts, and never lowers it by more than O(log log n). The same machinery applies to iterated remainder sets, tying their growth to a long-standing open problem about the length of Pierce expansions.","feed_headline":"Distinct remainders settle at 0.2296 per n","feed_subtitle":"A rigorous proof confirms the long-suspected density and bounds how the count jumps from n to n+1.","key_machinery":"The mechanism is a least-prime-factor sieve. Lemma 1 recasts membership of a remainder r in S(n) as a divisibility condition on n−r, namely that n−r has a proper divisor at least r+1; this is equivalent to n−r having a prime factor p ≤ (n−r)/(r+1). Counting remainders then reduces to counting integers m = n−r in short intervals that are divisible by p but no smaller prime, via inclusion–exclusion sets D_p with error O(2^{π(p−1)}). The error term of Theorem 1 comes from choosing the cutoff X = log n and bounding the tail by Σ_{p≥X} 1/p². For differences, the 'not transferred' set T(n, n+1) is characterized by r+1 being the largest proper divisor of n−r with prime cofactor, and for odd n the p","core_discovery":"The paper's central claim is that the number s(n) of distinct remainders n mod k for 1 ≤ k ≤ floor(n/2) has an exact linear asymptotic density: s(n) = c·n + O(n/(log n log log n)), where c = Σ_p 1/(p(p+1)) · Π_{q<p}(1 − 1/q) ≈ 0.2296. This confirms, with a rigorous error term, a density that had previously only been conjectured from numerical data. The paper also proves that s(n+1) is never more than one larger than s(n), that arbitrarily large downward jumps occur, and that every downward jump is at most O(log log n) in size. For the iterated remainder sets S_j(n), it establishes density bounds 1/(j+2)! ≤ liminf s_j(n)/n and limsup s_j(n)/n ≤ 1/(j+2), tying the problem to the length of Pier","pith_inferences":["The same interval-counting sieve should give an asymptotic for remainders in other ranges, such as 1 ≤ k ≤ α n for fixed α, with a density that varies continuously in α.","The colour-coded plots for S2(n) and S3(n) suggest the limiting density depends on n modulo small primes; proving Problem 2 would amount to exhibiting two subsequences of n with different limiting densities.","Lemma 8 suggests a way to predict large downward jumps: they are produced by multiplying n+1 by a new large prime, as in the recursive construction of Proposition 4, so one can test whether the first drop of size k always occurs for n equal to a product of initial primes minus 1."],"forward_implications":["The exact density of the remainder-count sequence is no longer a conjecture: any large-n computation of s(n) can be compared against c·n with a rigorous error bound.","Because increases are bounded by one while decreases are unbounded, the sequence s(n) has a one-sided Lipschitz character; consecutive-value drops of any fixed size occur infinitely often, yet never faster than log log n at scale n.","The upper bound max S_j(n) ≤ n/(j+2) gives a simple universal shrinkage rate for iterated remainder sets, independent of arithmetic properties of n.","The lower bound in Theorem 3 supplies a linear-in-n growth rate for iterated remainder sets, which is the same scale as the known upper bounds on Pierce-expansion length.","The paper leaves the error term open; if the numerical O(n^{1/3}) suggestion is correct, it would imply s(n) is far more regular than the sieve proof alone guarantees."],"supporting_citations":[{"why":"The StackExchange post that conjectured the constant c and sketched the density heuristic; Theorem 1 turns this into a rigorous asymptotic.","marker":"[4]"},{"why":"Rosser and Schoenfeld's explicit bound π(n) ≤ 1.3 n/log n controls the number of inclusion–exclusion terms in Lemma 3 and hence the error term.","marker":"[7]"},{"why":"Shallit's work on Pierce expansions defines the P(n) length problem that the iterated remainder sets S_j(n) are connected to via Lemma 10.","marker":"[9]"},{"why":"Erdős and Shallit's bounds on the length of finite Pierce and Engel series are the prior result (13) that the iterated-set bounds build on.","marker":"[2]"},{"why":"Chase and Pandey's improved upper/lower bounds on P(n) are quoted in (13) and motivate the comparison with s_j(n).","marker":"[1]"}],"fun_headline_variants":["Remainders settle at constant density c≈0.2296","Proof fixes density of distinct remainders at 0.2296","Bounding jumps in remainders: down jumps unbounded but slow","Iterated remainder sets sized: from factorial to polynomial bounds","Exact density for distinct remainders, with tight error term"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The lower-bound proof for iterated remainder sets assumes, in its base case, that S0(n) contains 0, although S0(n) is defined as {1, …, floor(n/2)}; starting the induction at r=1 repairs the argument, but the written proof depends on the uncorrected statement.","fun_headline_variants_meta":{"raw":{"variants":["Remainders settle at constant density c≈0.2296","Proof fixes density of distinct remainders at 0.2296","Bounding jumps in remainders: down jumps unbounded but slow","Iterated remainder sets sized: from factorial to polynomial bounds","Exact density for distinct remainders, with tight error term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000508,"raw_usage":{"total_tokens":2315,"prompt_tokens":753,"completion_tokens":1562,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":1474}},"tokens_in":497,"tokens_out":1562,"duration_ms":11438,"temperature":1.0,"reasoning_tokens":1474,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:46:02.418027+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify Theorem 1 by exact enumeration: if the normalized error (s(n) − c·n)/(n/(log n log log n)) is unbounded over n up to, say, 10^8, the claimed O-term fails; the paper's own data to 10^7 already fits a much smaller error. For the iterated lower bound, the literal base case of Lemma 13 fails at n=4: the proof requires {0,1} ⊆ S0(4), but S0(4) = {1,2}.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The StackExchange post that conjectured the constant c and sketched the density heuristic; Theorem 1 turns this into a rigorous asymptotic."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shallit's work on Pierce expansions defines the P(n) length problem that the iterated remainder sets S_j(n) are connected to via Lemma 10."},{"cited_title":"Erd˝ os and J","cited_arxiv_id":null,"evidence_quote":"Erdős and Shallit's bounds on the length of finite Pierce and Engel series are the prior result (13) that the iterated-set bounds build on."},{"cited_title":"On the length of Pierce expansions","cited_arxiv_id":"2211.08374","evidence_quote":"Chase and Pandey's improved upper/lower bounds on P(n) are quoted in (13) and motivate the comparison with s_j(n)."}],"review_version":1}