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 A329706 #14 May 25 2021 08:07:20
%S A329706 1,3,597,609,1791,2035,3403,3701,4263,27515,27955
%N A329706 Odd numbers k such that Sum_{j=1..(k-1)/2, gcd(j,k)=1} 1/j == -2*q_2(k) + k*q_2(k)^2 (mod k^3), where q_2(k) = (2^phi(k) - 1)/k is the Euler quotient of k to base 2.
%C A329706 Emma Lehmer proved that Sum_{j=1..(p-1)/2} 1/j == -2*q_2(p) + p*q_2(p)^2 (mod p^2) for all odd primes p.
%C A329706 Tianxin Cai generalized Lehmer's congruence and proved that Sum_{j=1..(k-1)/2, gcd(j,k)=1} 1/j == -2*q_2(k) + k*q_2(k)^2 (mod k^2) for all odd numbers k.
%C A329706 This sequence includes the odd numbers k for which the congruence is still valid when (mod k^2) is being replaced with (mod k^3).
%C A329706 The prime terms are 3, 3701, ...
%C A329706 No more terms below 147000.
%H A329706 Tianxin Cai, <a href="https://doi.org/10.4064/aa103-4-1">A congruence involving the quotients of Euler and its applications (I)</a>, Acta Arithmetica, Vol. 103, No. 4 (2002), pp. 313-320.
%H A329706 Tianxin Cai, <a href="https://doi.org/10.1007/978-1-4757-3621-2_5">A Generalization of E. Lehmer's Congruence and Its Applications</a>, in: ChaohuaJia and Kohji Matsumoto (eds.), Analytic Number Theory, Springer, Boston, MA, 2002, pp. 93-98.
%H A329706 Emma Lehmer, <a href="https://doi.org/10.2307/1968791">On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson</a>, Annals of Mathematics, Second Series, Vol. 39, No. 2 (1938), pp. 350-360, <a href="https://web.archive.org/web/20190509083052/http://gradelle.educanet2.ch/christian.aebi/.ws_gen/14/Emma_Lehmer_1938.pdf">alternative link</a>.
%t A329706 q[n_] := (2^EulerPhi[n] - 1)/n; Select[Range[1, 2100, 2], Divisible[Numerator[Sum[Boole @ CoprimeQ[j, #]/j, {j, 1, (# - 1)/2}] + 2*q[#] - #*q[#]^2], #^3] &]
%Y A329706 Cf. A000010, A001226.
%K A329706 nonn,more
%O A329706 1,2
%A A329706 _Amiram Eldar_, Feb 28 2020