A371732 Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 288, 512, 576, 1024, 2048, 3072, 4096, 8192, 16384, 32768, 32800, 33024, 33056, 65536, 65600, 66048, 66112, 131072, 132096, 133120, 134144, 262144, 266240, 524288, 528384, 786432, 790528, 1048576, 1056768, 2097152

Offset: 1

Author

Gus Wiseman, Apr 13 2024

Keywords

• nonn
• base

Examples

			The terms together with their binary expansions and binary indices begin:
        1:                1 ~ {1}
        2:               10 ~ {2}
        4:              100 ~ {3}
        8:             1000 ~ {4}
       12:             1100 ~ {3,4}
       16:            10000 ~ {5}
       32:           100000 ~ {6}
       64:          1000000 ~ {7}
      128:         10000000 ~ {8}
      144:         10010000 ~ {5,8}
      256:        100000000 ~ {9}
      288:        100100000 ~ {6,9}
      512:       1000000000 ~ {10}
      576:       1001000000 ~ {7,10}
     1024:      10000000000 ~ {11}
     2048:     100000000000 ~ {12}
     3072:     110000000000 ~ {11,12}
     4096:    1000000000000 ~ {13}
     8192:   10000000000000 ~ {14}
    16384:  100000000000000 ~ {15}
    32768: 1000000000000000 ~ {16}
    32800: 1000000000100000 ~ {6,16}
		

Crossrefs

For prime instead of binary indices we have A326534.

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

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

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

A321142 and A371794 count non-biquanimous strict partitions.

A321452 counts quanimous partitions, ranks A321454.

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

A357976 ranks the biquanimous partitions counted by A002219 aerated.

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

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

Programs

Mathematica

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SameQ@@Total/@bix/@bix[#]&]