This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377570 #33 Apr 15 2025 00:24:55 %S A377570 1,3,5,7,8,12,12,16,17,21,19,28,23,29,30,35,30,42,34,46,43,46,41,61, %T A377570 48,55,55,65,53,76,57,74,68,73,71,94,69,82,81,100,77,106,81,103,101, %U A377570 100,89,129,97,116,107,122,102,136,114,139,121,127,114,169,118,137 %N A377570 a(n) = round((H(n) + e^H(n)*log(H(n)) + sigma(n))/2). %C A377570 The idea of this sequence is finding the nearest integer to the arithmetic mean between the two sides of the inequality that is equivalent to the Riemann hypothesis. The inequality is sigma(n) < H(n) + e^H(n)*log(H(n)), where H(n) are the harmonic numbers. %C A377570 Conjecture: For each positive integer k, there exist at least one positive integer m such that there are exactly k terms in this sequence that are equal to m. In the first 10000 terms, each value occurs at most 4 times. However, as n becomes larger, we can see from the scatterplot that the conjecture might be true. - _Ahmad J. Masad_, Apr 03 2025 %H A377570 Amiram Eldar, <a href="/A377570/b377570.txt">Table of n, a(n) for n = 1..10000</a> %e A377570 For n=6, sigma(6)=12 and H(6)=1/1+1/2+1/3+1/4+1/5+1/6=2.45 and (2.45+e^2.45*log(2.45)+12)/2=12.417... and round(12.417...)=12; hence a(6)=12. %t A377570 f[x_] := x + Exp[x] * Log[x]; a[n_] := Round[(f[HarmonicNumber[n]] + DivisorSigma[1, n])/2]; Array[a, 100] (* _Amiram Eldar_, Nov 01 2024 *) %Y A377570 Cf. A000203, A001008/A002805, A057641. %K A377570 nonn %O A377570 1,2 %A A377570 _Ahmad J. Masad_, Nov 01 2024