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A328379 a(n) is the sum of the distinct numbers whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

%I A328379 #92 Dec 02 2019 03:46:31
%S A328379 0,1,3,4,7,11,12,11,15,24,31,29,28,37,33,26,31,49,66,61,71,92,85,67,
%T A328379 60,87,103,90,77,95,78,57,63,98,133,121,150,191,177,138,151,215,254,
%U A328379 219,197,240,199,145,124,185,237,210,235,293,262,199,165,230,263,223
%N A328379 a(n) is the sum of the distinct numbers whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.
%H A328379 Rémy Sigrist, <a href="/A328379/b328379.txt">Table of n, a(n) for n = 0..16384</a>
%H A328379 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A328379 A078823(n) <= a(n).
%F A328379 a(2^k) = 2^(k+1)-1 for any k >= 0.
%F A328379 a(2^k-1) = A000295(k+1) for any k >= 0.
%e A328379 The first terms, alongside the binary representations of n as well as those of the numbers that appear in it, are:
%e A328379   n   a(n)  bin(n)  {bin(s)}
%e A328379   --  ----  ------  ----------------------------
%e A328379    0     0       0  {0}
%e A328379    1     1       1  {1}
%e A328379    2     3      10  {0, 1, 10}
%e A328379    3     4      11  {1, 11}
%e A328379    4     7     100  {0, 1, 10, 100}
%e A328379    5    11     101  {0, 1, 10, 11, 101}
%e A328379    6    12     110  {0, 1, 10, 11, 110}
%e A328379    7    11     111  {1, 11, 111}
%e A328379    8    15    1000  {0, 1, 10, 100, 1000}
%e A328379    9    24    1001  {0, 1, 10, 11, 100, 101, 1001}
%e A328379   10    31    1010  {0, 1, 10, 11, 100, 101, 110, 1010}
%o A328379 (PARI) a(n, base=2) = { my (b=digits(n, base), s=[0]); for (k=1, #b, s = setunion(s, apply(o -> base*o+b[k], s))); vecsum(s) }
%Y A328379 Cf. A000295, A078823, A329873.
%K A328379 nonn,base
%O A328379 0,3
%A A328379 _Rémy Sigrist_, Nov 30 2019