Two corrections to §8 of Shiu (2016): E_2113 contains 1364, and the last run at 10^4 has 156 members

Section 8 of J. Shiu, "Denominators of harmonic numbers" (arXiv:1607.02863v2), contains two typographical errors. (i) Table 2 lists E_2113 = {44, 443, 553, 748, 1384, 1559, 1669, 2068, 2112}. Shiu himself proves that E_p is invariant under the involution m -> p-1-m, for 1 <= m <= p-2. For p = 2113 this forces 1364 (since 748+1364+1 = 2113), whereas the printed 1384 would force 728. Direct enumeration confirms 1364 is in E_2113 and 728 is not. The entry 1384 should read 1364; the cardinality |E_2113| = 9 is unchanged. (ii) The last of the 26 maximal runs of good n up to 10^4 is printed as 9156_155; the printed run lengths sum to 2640, whereas the text states that there are 2641 such n. Exact computation gives 26 runs summing to 2641, the last being 9156 <= n <= 9311 with 156 members. This second point has already been recorded in the public research notes at erdosproblems.ai (problem 291); it is included here for completeness and no priority is claimed for it. Reproducible code (Python standard library only) is contained in the attached document.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-10
DOI
https://doi.org/10.5281/zenodo.24098146
Primary Topic
Analytic Number Theory Research
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article
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article

Two corrections to §8 of Shiu (2016): E_2113 contains 1364, and the last run at 10^4 has 156 members

Yuanye Tian
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
article

Two corrections to §8 of Shiu (2016): E_2113 contains 1364, and the last run at 10^4 has 156 members

Yuanye Tian
article en

Abstract

Section 8 of J. Shiu, "Denominators of harmonic numbers" (arXiv:1607.02863v2), contains two typographical errors. (i) Table 2 lists E_2113 = {44, 443, 553, 748, 1384, 1559, 1669, 2068, 2112}. Shiu himself proves that E_p is invariant under the involution m -> p-1-m, for 1 <= m <= p-2. For p = 2113 this forces 1364 (since 748+1364+1 = 2113), whereas the printed 1384 would force 728. Direct enumeration confirms 1364 is in E_2113 and 728 is not. The entry 1384 should read 1364; the cardinality |E_2113| = 9 is unchanged. (ii) The last of the 26 maximal runs of good n up to 10^4 is printed as 9156_155; the printed run lengths sum to 2640, whereas the text states that there are 2641 such n. Exact computation gives 26 runs summing to 2641, the last being 9156 <= n <= 9311 with 156 members. This second point has already been recorded in the public research notes at erdosproblems.ai (problem 291); it is included here for completeness and no priority is claimed for it. Reproducible code (Python standard library only) is contained in the attached document.

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
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