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 A051404 #58 Aug 26 2025 04:55:50
%S A051404 1,2,3,4,6,9,10,12,18,33,34,36,40,64,66,192,256,264,272,513,514,516,
%T A051404 576,768,1026,1056,2304,16392,65664,81920,532480,545259520
%N A051404 Numbers k such that neither 4 nor 9 divides binomial(2k-1,k) (almost certainly finite).
%C A051404 Complete up to 2^64 = 18446744073709551616.
%C A051404 Complete up to 2^30000. - _Don Reble_, Oct 27 2013
%C A051404 A number n is in the sequence if and only if the following inequalities hold s_2(n) <= 2 and s_3(n) + s_3(n-1) - s_3(2*n-1) <= 2, where s_m(n) is sum of digits of n in base m. - _Vladimir Shevelev_, Oct 30 2013
%C A051404 Equivalently, a number n is in the sequence if and only if there is at most 1 "carry" when adding n and n-1 in both base-2 arithmetic and base-3 arithmetic. - _Tom Edgar_, Oct 31 2013
%D A051404 Adrien-Marie Legendre, Théorie de Nombres, Firmin Didot Frères, Paris, 1830.
%H A051404 E. E. Kummer, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002147432">Uber die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen</a>, J. Reine Angew Math. 44 (1852), 93-146.
%H A051404 Don Reble, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-October/011806.html">A051404</a>, SeqFan Post, Oct 30 2013.
%H A051404 Vladimir Shevelev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Shevelev/shevelev14.html">Binomial coefficient predictors</a>, J. of Integer Sequences, Vol. 14 (2011), Article 11.2.8.
%H A051404 Vladimir Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-October/011817.html">Re: A051404</a>, SeqFan Post, Oct 30 2013.
%H A051404 Wikipedia, <a href="http://en.wikipedia.org/wiki/Kummer's_theorem">Kummer's Theorem</a>.
%e A051404 For n = 64 we have s_2(64) = 1, s_3(n) = 4, s_3(64-1) = 3, s_3(2*64-1) = 5 and 4+3-5 = 2. So 64 is in the sequence. - _Vladimir Shevelev_, Oct 30 2013
%t A051404 s[n_] :=DigitSum[n, 3]; With[{emax = 30}, Select[Flatten@ Table[2^e1 + If[e2 < 0, 0, 2^e2], {e1, 0, emax}, {e2, -1, e1-1}], s[#] + s[#-1] - s[2*#-1] <= 2 &]] (* _Amiram Eldar_, Aug 26 2025 *)
%o A051404 (PARI) isok(k) = my(b=binomial(2*k-1,k)); (b%4) && (b%9); \\ _Michel Marcus_, Jan 22 2025
%o A051404 (PARI) s(n) = sumdigits(n, 3);
%o A051404 list(emax = 30) = {my(k); for(e1 = 0, emax, for(e2 = -1, e1-1, k = 1 << e1 + if(e2 >= 0, 1 << e2); if(s(k) + s(k-1) - s(2*k-1) <= 2, print1(k, ", "))));} \\ _Amiram Eldar_, Aug 26 2025
%Y A051404 Cf. A000120, A001700, A053735, A110556.
%K A051404 nonn,changed
%O A051404 1,2
%A A051404 _David W. Wilson_