A367913 - OEIS

%I #6 Dec 17 2023 11:23:32

%S 1,4,64,20,68,320,52,84,16448,324,832,116,1104,308,816,340,836,848,

%T 1108,1136,1360,3152,16708,372,5188,5216,852,880,2884,1364,13376,1392,

%U 3184,3424,17220,5204,5220,2868,5728,884,19536,66896,2900,1396,21572,3188,3412

%N Least number k such that there are exactly n ways to choose a multiset consisting of a binary index of each binary index of k.

%C A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

%e The terms together with the corresponding set-systems begin:

%e       1: {{1}}

%e       4: {{1,2}}

%e      64: {{1,2,3}}

%e      20: {{1,2},{1,3}}

%e      68: {{1,2},{1,2,3}}

%e     320: {{1,2,3},{1,4}}

%e      52: {{1,2},{1,3},{2,3}}

%e      84: {{1,2},{1,3},{1,2,3}}

%e   16448: {{1,2,3},{1,2,3,4}}

%e     324: {{1,2},{1,2,3},{1,4}}

%e     832: {{1,2,3},{1,4},{2,4}}

%e     116: {{1,2},{1,3},{2,3},{1,2,3}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];

%t c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,1000}];

%t Table[Position[c,n][[1,1]],{n,spnm[c]}]

%Y A version for multisets and divisors is A355734.

%Y With distinctness we have A367910, firsts of A367905, sorted A367911.

%Y Positions of first appearances in A367912.

%Y The sorted version is A367915.

%Y For sequences we have A368111, firsts of A368109, sorted A368112.

%Y For sets we have A368184, firsts of A368183, sorted A368185.

%Y A048793 lists binary indices, length A000120, sum A029931.

%Y A058891 counts set-systems, covering A003465, connected A323818.

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355741, A355744.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 16 2023