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 A084699 #62 Dec 08 2023 06:21:52
%S A084699 12,30,56,424,992,16256,58288,119984,356992,1194649,9973504,12327121,
%T A084699 13141696,22891184,67100672,233850649
%N A084699 Composite integers j such that binomial(2*j,j) == 2^j (mod j).
%C A084699 If p is prime, binomial(2*p,p) == 2^p (mod p).
%C A084699 a(17) > 10^9.
%C A084699 From Gabriel Guedes and _Ricardo Machado_, Nov 16 2023: (Start)
%C A084699 Theorem. Let j = (2^k)*p, where p is an odd prime and k is in N; then binomial(2*j,j) == 2^j (mod j) if and only if p satisfies the following conditions:
%C A084699 a) p divides binomial(2^(k+1),2^k) - 2^(2^k);
%C A084699 b) p has at least k 1's in its binary expansion.
%C A084699 Theorem. If m is an even perfect number then j = 2m satisfies the congruence binomial(2*j,j) == 2^j (mod j). See A000396.
%C A084699 Theorem. Let j = p^2 with p a prime number. Then p is a Wieferich prime if and only if binomial(2*j,j) == 2^j (mod j). See A001220. (End)
%C A084699 Contains 17179738112 and 274877382656 (from Guedes-Machado paper). - _Michael De Vlieger_, Nov 22 2023
%C A084699 Contains 3386741824, 750984028672, 33029195197184, 1145067923695616, 422612863956511744. - _Ricardo Machado_, Nov 23 2023
%C A084699 Contains 84385517065596416, 62648180117928433664, 273984397779878971648, 36506097537257040703232. - _Max Alekseyev_, Dec 07 2023
%H A084699 Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI/GP Scripts for Miscellaneous Math Problems</a>, sect. III: Binomial coefficients modulo integers, binomod.gp (V. 1.4, 11/2015).
%H A084699 Gabriel Guedes and Ricardo Machado, <a href="https://nntdm.net/volume-29-2023/number-4/705-712/">Perfect numbers, Wieferich primes and the solutions of binomial(2n, n) congruent to 2^n mod n</a>, Notes Num. Theor. Disc. Math. (NNTDM 2023) Vol. 29, No. 4, 705-712.
%o A084699 (PARI) lista(nn) = {forcomposite(n=1, nn, if (binomod(2*n, n, n) == Mod(2, n)^n, print1(n, ", ")));} \\ _Michel Marcus_, Dec 06 2013 and Dec 03 2023
%Y A084699 Contains A139256 as a subsequence.
%Y A084699 Cf. A015910, A059288, A080469, A082180, A109760, A109769, A228562, A328497.
%K A084699 nonn,more
%O A084699 1,1
%A A084699 _Benoit Cloitre_, Oct 15 2003
%E A084699 More terms from _David Wasserman_, Jan 03 2005
%E A084699 a(11)-a(16) from _Max Alekseyev_, Aug 05 2011