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Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).
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%I #7 Apr 14 2024 03:49:39

%S 1,2,4,8,12,16,32,64,128,144,256,288,512,576,1024,2048,3072,4096,8192,

%T 16384,32768,32800,33024,33056,65536,65600,66048,66112,131072,132096,

%U 133120,134144,262144,266240,524288,528384,786432,790528,1048576,1056768,2097152

%N Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

%e The terms together with their binary expansions and binary indices begin:

%e 1: 1 ~ {1}

%e 2: 10 ~ {2}

%e 4: 100 ~ {3}

%e 8: 1000 ~ {4}

%e 12: 1100 ~ {3,4}

%e 16: 10000 ~ {5}

%e 32: 100000 ~ {6}

%e 64: 1000000 ~ {7}

%e 128: 10000000 ~ {8}

%e 144: 10010000 ~ {5,8}

%e 256: 100000000 ~ {9}

%e 288: 100100000 ~ {6,9}

%e 512: 1000000000 ~ {10}

%e 576: 1001000000 ~ {7,10}

%e 1024: 10000000000 ~ {11}

%e 2048: 100000000000 ~ {12}

%e 3072: 110000000000 ~ {11,12}

%e 4096: 1000000000000 ~ {13}

%e 8192: 10000000000000 ~ {14}

%e 16384: 100000000000000 ~ {15}

%e 32768: 1000000000000000 ~ {16}

%e 32800: 1000000000100000 ~ {6,16}

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

%t Select[Range[1000],SameQ@@Total/@bix/@bix[#]&]

%Y For prime instead of binary indices we have A326534.

%Y A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

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

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y A321142 and A371794 count non-biquanimous strict partitions.

%Y A321452 counts quanimous partitions, ranks A321454.

%Y A326031 gives weight of the set-system with BII-number n.

%Y A357976 ranks the biquanimous partitions counted by A002219 aerated.

%Y A371731 ranks the non-biquanimous partitions counted by A371795, A006827.

%Y Cf. A035470, A038041, A237258, A320324, A321453, A321455, A326518, A336137, A371783, A371791, A371796.

%K nonn,base

%O 1,2

%A _Gus Wiseman_, Apr 13 2024