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 A370640 #13 Mar 28 2025 15:21:11
%S A370640 1,1,1,3,3,8,17,32,32,77,144,242,383,580,843,1201,1201,2694,4614,7096,
%T A370640 10219,14186,19070,25207,32791,42160,53329,66993,82811,101963,124381,
%U A370640 151286,151286,324695,526866,764438,1038089,1358129,1725921,2154668,2640365,3202985
%N A370640 Number of maximal subsets of {1..n} such that it is possible to choose a different binary index of each element.
%C A370640 A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%C A370640 Also choices of A070939(n) elements of {1..n} such that it is possible to choose a different binary index of each.
%e A370640 The a(0) = 1 through a(6) = 17 subsets:
%e A370640 {} {1} {1,2} {1,2} {1,2,4} {1,2,4} {1,2,4}
%e A370640 {1,3} {1,3,4} {1,2,5} {1,2,5}
%e A370640 {2,3} {2,3,4} {1,3,4} {1,2,6}
%e A370640 {1,3,5} {1,3,4}
%e A370640 {2,3,4} {1,3,5}
%e A370640 {2,3,5} {1,3,6}
%e A370640 {2,4,5} {1,4,6}
%e A370640 {3,4,5} {1,5,6}
%e A370640 {2,3,4}
%e A370640 {2,3,5}
%e A370640 {2,3,6}
%e A370640 {2,4,5}
%e A370640 {2,5,6}
%e A370640 {3,4,5}
%e A370640 {3,4,6}
%e A370640 {3,5,6}
%e A370640 {4,5,6}
%e A370640 The a(0) = 1 through a(6) = 17 set-systems:
%e A370640 {1} {1}{2} {1}{2} {1}{2}{3} {1}{2}{3} {1}{2}{3}
%e A370640 {1}{12} {1}{12}{3} {1}{12}{3} {1}{12}{3}
%e A370640 {2}{12} {2}{12}{3} {1}{2}{13} {1}{2}{13}
%e A370640 {2}{12}{3} {1}{2}{23}
%e A370640 {2}{3}{13} {1}{3}{23}
%e A370640 {1}{12}{13} {2}{12}{3}
%e A370640 {12}{3}{13} {2}{3}{13}
%e A370640 {2}{12}{13} {1}{12}{13}
%e A370640 {1}{12}{23}
%e A370640 {1}{13}{23}
%e A370640 {12}{3}{13}
%e A370640 {12}{3}{23}
%e A370640 {2}{12}{13}
%e A370640 {2}{12}{23}
%e A370640 {2}{13}{23}
%e A370640 {3}{13}{23}
%e A370640 {12}{13}{23}
%t A370640 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A370640 Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]
%o A370640 (PARI) lista(nn) = my(b, m=Map(Mat([[[]], 1])), t, u, v, w, z); for(n=0, nn, t=Mat(m)~; b=Vecrev(binary(n)); u=select(i->b[i], [1..#b]); for(i=1, #t, v=t[1, i]; w=List([]); for(j=1, #v, for(k=1, #u, if(!setsearch(v[j], u[k]), listput(w, setunion(v[j], [u[k]]))))); w=Set(w); if(#w, z=0; mapisdefined(m, w, &z); mapput(m, w, z+t[2, i]))); print1(mapget(m, [[1..#b]]), ", ")); \\ _Jinyuan Wang_, Mar 28 2025
%Y A370640 Dominated by A357812.
%Y A370640 The version for set-systems is A368601, max of A367902 (complement A367903).
%Y A370640 For prime indices we have A370585, with n A370590, see also A370591.
%Y A370640 This is the maximal case of A370636 (complement A370637).
%Y A370640 The case of a unique choice is A370638.
%Y A370640 The case containing n is A370641, non-maximal A370639.
%Y A370640 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A370640 A058891 counts set-systems, A003465 covering, A323818 connected.
%Y A370640 A070939 gives length of binary expansion.
%Y A370640 A096111 gives product of binary indices.
%Y A370640 A307984 counts Q-bases of logarithms of positive integers.
%Y A370640 A355741 counts choices of a prime factor of each prime index.
%Y A370640 Cf. A133686, A326031, A326702, A367905, A367909, A367912, A368109, A368110, A370592, A370642.
%K A370640 nonn
%O A370640 0,4
%A A370640 _Gus Wiseman_, Mar 10 2024
%E A370640 More terms from _Jinyuan Wang_, Mar 28 2025