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Melfi 2004 certain positive integer sequences
conjecture_1: Melfi's conjecture that for s >= 1 and a pairwise coprime sequence A of integers at least 2 with sum 1/log a above the printed threshold log 2, the subset sums of Pow(A;s) have positive lower asymptotic density; the printed threshold admits the single base 3, whose subset sums have density zero.
proposition_1: Melfi's counterexample to the only-if half of the Burr, Erdos, Graham and Li conjecture: for every epsilon > 0 there is an infinite set A of integers at least 2 with sum 1/(a-1) < epsilon such that Pow(A;s) is complete for every s >= 1.
theorem_3: Melfi's lower bound p_{(2,1,2)}(n) >> n^{0.025} for the counting function of the (2,1,2)-numbers, the positive integers whose binary digit sum equals that of their square.
theorem_4: Melfi's lower bound p_{(2,2,2)}(n) >> n^{0.0909} for the counting function of the (2,2,2)-numbers, the positive integers n whose square has binary digit sum twice that of n.
Melfi, Giuseppe, On certain positive integer sequences. Riv. Mat. Univ. Parma (7) 3* (2004), 253--260.
A survey based on a talk at the Second Italian Meeting of Number Theory (Parma, November 2003), with four topical sections and some new results. Section 2 (pp. 254--255) reviews practical numbers: Stewart's characterization; Saias's Chebyshev-type bounds c_1 x/log x < P(x) < c_2 x/log x for suitable constants (Theorem 1, p. 254); the author's proof that every even positive integer is a sum of two practical numbers; and the author's lower bound P_2(x) > x/exp(k (log x)^{1/2}), for each k > 2 + log(3/2) and all sufficiently large x, P_2 counting the practical m <= x with m + 2 practical (Theorem 2, p. 255). Section 3 (pp. 255--256) surveys sum-free sequences, those in which no term is a sum of distinct smaller terms: Erdos's results that such a sequence has density zero and that sum_j 1/n_j < 103, the constructions of Deshouillers, Erdos and the author with n_k ~ k^{3+delta} and of Luczak and Schoen with n_k ~ k^{2+delta}, and the bounds 2.064 < R < 4 on the supremum R of sum_j 1/n_j (Abbott; Levine and O'Sullivan). Section 4 (pp. 256--257) concerns complete sequences of powers: Proposition 1 constructs, for every eps > 0, an infinite set A of integers >= 2 with sum_{a in A} 1/(a-1) < eps such that Pow(A;s) is complete for every s >= 1, disproving the 'only if' half of the Burr-Erdos-Graham-Li conjecture for infinite A (the paper says the finite case is open); the section then reports Erdos's request for a proof that n_k << k, where n_1 < n_2 < ... are the positive integers that are sums of distinct powers of 3 and of 4, with n_k << k^{1.0353} as the best known result, and poses Conjecture 1 on positive lower density of subset sums of powers of pairwise coprime bases. Section 5 (pp. 257--259) defines (k,l,m)-numbers (Definition 1, p. 258), proves the lower bounds p_{(2,1,2)}(n) >> n^{0.025} (Theorem 3, p. 258) and p_{(2,2,2)}(n) >> n^{0.0909} (Theorem 4, p. 259), reports Sandor's announced upper bound p_{(2,1,2)}(n) << n^{0.9183}, and states two heuristic conjectures on these counting functions (Conjectures 2 and 3, p. 259).
Source: http://www.rivmat.unipr.it/vols/2004-3s/indice.html. No notice is printed in the file, and the journal's volume index page named here states no copyright or license term (http://www.rivmat.unipr.it/vols/2004-3s/indice.html, read 2026-10-02); the term is unstated.
Read status. Claims checked: Proposition 1 with its proof, Conjecture 1 with the remark after it, Definition 1 and Theorems 3 and 4 were read clause by clause on the printed pages. Theorems 3 and 4 are proved only in outline here; the full proofs, in the author's preprint arXiv:math/0402458, are not checked. Theorems 1 and 2 and the surveyed results of Sections 2 and 3 are other papers' results, reported here and not transcribed.
Bears on.
- #124: background only. Proposition 1 shows that the reciprocal-sum condition sum 1/(a-1) >= 1 is not necessary for completeness when the base set is infinite; the problem asks whether that condition is sufficient for finite tuples of bases (with gcd 1 as well when the powers start at a positive exponent), and the proposition decides no instance of it.
- #125: the paper reports Erdos's question whether n_k << k for the sums of distinct powers of 3 and of 4, a set contained in the problem's sumset A + B, and the bound n_k << k^{1.0353} from the author's 2001 paper; it proves nothing new on the problem. Conjecture 1, applied to the bases {3,4}, would give A + B positive lower density; the conjecture's page explains why its printed threshold needs correcting and how it stands against the disproof the problem page records.
- #876: background only. Section 3 reports, without proof, other papers' results on the growth of sum-free sequences in the problem's sense; it adds no result of its own.
Results. Proposition 1 (p. 256); Conjecture 1 (p. 257); Theorem 3 (p. 258, with Definition 1); Theorem 4 (p. 259).
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.