cp's OEIS Frontend

A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

%I A385575 #10 Jul 18 2025 22:02:46
%S A385575 1,2,4,8,11,13,16,19,22,25,26,32,35,38,44,49,50,52,64,67,70,76,87,88,
%T A385575 91,93,97,98,100,104,107,109,117,128,131,134,140,151,152,155,157,167,
%U A385575 174,176,179,182,185,186,193,194,196,200,203,205,208,211,214,217
%N A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.
%C A385575 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.
%e A385575 The terms together with their binary expansions and binary indices begin:
%e A385575     1:       1 ~ {1}
%e A385575     2:      10 ~ {2}
%e A385575     4:     100 ~ {3}
%e A385575     8:    1000 ~ {4}
%e A385575    11:    1011 ~ {1,2,4}
%e A385575    13:    1101 ~ {1,3,4}
%e A385575    16:   10000 ~ {5}
%e A385575    19:   10011 ~ {1,2,5}
%e A385575    22:   10110 ~ {2,3,5}
%e A385575    25:   11001 ~ {1,4,5}
%e A385575    26:   11010 ~ {2,4,5}
%e A385575    32:  100000 ~ {6}
%e A385575    35:  100011 ~ {1,2,6}
%e A385575    38:  100110 ~ {2,3,6}
%e A385575    44:  101100 ~ {3,4,6}
%e A385575    49:  110001 ~ {1,5,6}
%e A385575    50:  110010 ~ {2,5,6}
%e A385575    52:  110100 ~ {3,5,6}
%e A385575    64: 1000000 ~ {7}
%e A385575    67: 1000011 ~ {1,2,7}
%e A385575    70: 1000110 ~ {2,3,7}
%e A385575    76: 1001100 ~ {3,4,7}
%e A385575    87: 1010111 ~ {1,2,3,5,7}
%e A385575    88: 1011000 ~ {4,5,7}
%e A385575    91: 1011011 ~ {1,2,4,5,7}
%e A385575    93: 1011101 ~ {1,3,4,5,7}
%e A385575    97: 1100001 ~ {1,6,7}
%e A385575    98: 1100010 ~ {2,6,7}
%e A385575   100: 1100100 ~ {3,6,7}
%t A385575 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A385575 Select[Range[100],Length[Split[bpe[#],#2==#1+1&]]==Length[Split[bpe[#],#2!=#1+1&]]&]
%o A385575 (PARI) is_ok(n)=hammingweight(n)==2*hammingweight(bitand(n,n>>1))+1 \\ _Christian Sievers_, Jul 18 2025
%Y A385575 The LHS rank statistic is A069010, counted by A034839 (for partitions A384881, A116674).
%Y A385575 The RHS rank statistic is A384890, counted by A384893 (for partitions A268193, A384905).
%Y A385575 Subsets of this type are counted by A385572, with n A217615.
%Y A385575 A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
%Y A385575 A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
%Y A385575 Cf. A000079, A010027, A053538, A210034, A384177, A384879, A384889.
%K A385575 nonn
%O A385575 1,2
%A A385575 _Gus Wiseman_, Jul 04 2025