Tang 2026 hofstadter consecutive sum sequence omits infinitely

../

lemma_2_1: A positive integer is a sum of at least two consecutive positive integers exactly when it is not a power of 2; one of the paper's two preliminary ingredients for its results on Problem 423.

theorem_1_3: For the Hofstadter consecutive-sum sequence a_n of Problem 423, the difference a_n - n is nondecreasing and unbounded, so a_n = n + omega(1) and the sequence omits infinitely many positive integers.

theorem_1_4: For the Hofstadter consecutive-sum sequence a_n of Problem 423, a_n is at least n + log log n / log 20 minus a bounded quantity.

theorem_1_5: For the Hofstadter consecutive-sum sequence a_n of Problem 423 and every epsilon > 0, a_n is at most a constant depending on epsilon times n to the power 4175/2506 + epsilon, for all n at least 1.

---

Quanyu Tang, The Hofstadter consecutive-sum sequence omits infinitely many positive integers. arXiv:2603.09939 (2026). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2603.09939), every other right reserved. The copy read for this card is version 2 (23 March 2026), and the labels below are its labels; version 1 (10 March 2026) states in its abstract only the lower bound a_n >= n + omega(1).

The paper studies the greedy self-generating sequence with a_1 = 1, a_2 = 2 and a_k the least integer above a_{k-1} expressible as a sum of at least two consecutive earlier terms (OEIS A005243), whose asymptotics Hofstadter asked about and which is problem 423. Theorem 1.3 shows that b_n = a_n - n is nondecreasing and unbounded, so a_n = n + omega(1) and the sequence omits infinitely many positive integers, settling the conjecture recorded in the OEIS comments. Theorem 1.4 makes this quantitative with a_n >= n + log log n / log 20 - O(1), and Theorem 1.5 gives a first polynomial upper bound a_n <= C_eps n^{4175/2506 + eps} for every eps > 0 and all n >= 1, whose proof joins the greedy structure of the sequence to a recent lower bound on the size of the difference set of a finite convex set. The preliminaries supply the two ingredients: Lemma 2.1, that a positive integer is a sum of at least two consecutive positive integers exactly when it is not a power of 2, and a finiteness consequence of the Schinzel-Tijdeman theorem (Corollary 2.3). Together the results give two-sided bounds toward Hofstadter's asymptotic question.

Source: https://arxiv.org/abs/2603.09939.

Bears on. #423: the problem asks for the asymptotic behavior of the sequence; Theorem 1.3 shows $a_n-n$ is nondecreasing and unbounded, and Theorems 1.4 and 1.5 bound $a_n$ between $n+\log\log n/\log20-O(1)$ and $C_\varepsilon n^{4175/2506+\varepsilon}$. None determines the asymptotics the problem asks for. The exponent $688/413$ in Section 6.2 (p. 14) is Sothanaphan's observation recorded in the paper, not one of its theorems.

Results.

• Theorem 1.3 (p. 2): $b_n=a_n-n$ is nondecreasing and unbounded, so $a_n=n+\omega(1)$ and the sequence omits infinitely many positive integers, settling the conjecture recorded in OEIS A005243.
• Theorem 1.4 (p. 2): $a_n\ge n+\log\log n/\log20-O(1)$.
• Theorem 1.5 (p. 2): for every $\varepsilon>0$ there is $C_\varepsilon>0$ with $a_n\le C_\varepsilon n^{4175/2506+\varepsilon}$ for all $n\ge1$.
• Lemma 2.1 (p. 2): a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of 2.

Together Theorems 1.4 and 1.5 give the abstract's $n+\Omega(\log\log n)\le a_n\ll n^{4175/2506+o(1)}$ (p. 1).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.