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 A308963 #52 Jan 17 2020 10:45:26
%S A308963 77,161,2261,12839,14231,18668831,1591100357
%N A308963 Lerch pseudoprimes: composite numbers m such that Sum_{k=1..m-1} k^{m-1} - (m-1)! == m (mod m^2).
%C A308963 According to Lerch's congruence (1905), if p is an odd prime, then Sum_{k=1..p-1} k^(p-1) - (p-1)! == p (mod p^2).
%C A308963 Equivalently, numbers m > 4 such that Sum_{k=1..m-1} k^(m-1) == m (mod m^2).
%C A308963 Equivalently, numbers m > 1 such that m*B_{m-1} == m (mod m^2), where B_k is the k-th Bernoulli number.
%C A308963 Equivalently, terms m of A121707 such that B_{m-1} == 1 (mod m).
%C A308963 Equivalently, numbers m > 1 such that A027641(m-1) == A027642(m-1) (mod m).
%C A308963 If m is a Lerch pseudoprime, then p-1 does not divide m-1 for every prime divisor p of m.
%C A308963 From _M. F. Hasler_, Jul 22 2019: (Start)
%C A308963 The Lerch primes A197632 satisfy Lerch's congruence "even" modulo p^3.
%C A308963 Up to a(7) all terms are either multiples of 7 or of 37, but not both. Will this pattern prevail?
%C A308963 We also note: a(1) = 7*11; a(2) = 7*(2*11 + 1) = a(1)/11*23; a(3) = 7*(2*7*23 + 1) = a(2)/23*17*19, a(5) = a(3)/17*107, i.e., a term in this subsequence has all but one of the prime factors of the preceding one. The subsequence (a(4), a(6), ...?) of terms divisible by 37 so far consists of semiprimes and therefore also has this property. (End)
%H A308963 Mathias Lerch, <a href="https://eudml.org/doc/158206">Zur Theorie des Fermatschen Quotienten (a^(p-1)-1)/p = q(a)</a>, Mathematische Annalen, Vol. 60, No. 4 (1905), pp. 471-490.
%H A308963 Jonathan Sondow, <a href="https://doi.org/10.1007/978-1-4939-1601-6_17">Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771</a>, In: Nathanson M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, Vol. 101, Springer, New York, NY, 2014, pp. 243-255. <a href="https://arxiv.org/abs/1110.3113">Preprint: arXiv:1110.3113 [math.NT]</a>.
%t A308963 s={}; Do[If[CompositeQ[n] && Mod[Sum[PowerMod[k, n-1, n^2], {k, 1, n-1}] - (n-1)! - n, n^2] == 0, AppendTo[s, n]],{n,1,2500}] ; s
%o A308963 (PARI) is_A308963(m)={sum(k=1,m-1,Mod(k,m^2)^(m-1))==m&&!isprime(m)&&m>4}
%o A308963 forcomposite(m=1,,is_A308963(m)&&print1(m",")) \\ Slow beyond 10000. - _M. F. Hasler_, Jul 22 2019
%Y A308963 A subsequence of A191677 and A121707.
%Y A308963 Cf. A027641, A027642, A197630, A197632.
%K A308963 nonn,more
%O A308963 1,1
%A A308963 _Amiram Eldar_ and _Thomas Ordowski_, Jul 03 2019
%E A308963 a(6)-a(7) from _Max Alekseyev_, Jul 09 2019