A369332 - OEIS

%I #17 Jan 29 2024 19:09:46

%S 1,17,186,12234,605714,30143621,865062888,374978871766,92420578210888,

%T 22764626902276757,4227156427366610576,1076625258046594762034,

%U 196829039855755478065982,34737980525681450161565604,3519580168264415862502129296,8186117385516870986118141242073

%N a(n) is the sum of numbers whose binary forms can be constructed using some or all of the binary digits of 1..n.

%e For a(3) = 186, the binary forms of n = 1, 2 and 3 are 1, 10 and 11. These together contain four 1's and one 0. The possible combinations to construct binary numbers of these are below with their equivalent decimal values:

%e        1     1

%e       10     2

%e       11     3

%e      101     5

%e      110     6

%e      111     7

%e     1011    11

%e     1101    13

%e     1110    14

%e     1111    15

%e    10111    23

%e    11011    27

%e    11101    29

%e    11110    30

%e            ---

%e    Total:  186

%o (PARI) a(n)={my(w=0,b=0); for(i=1, n, w+=hammingweight(i); b+=logint(i,2)+1); sum(j=0, w-1, sum(k=0, b-w, my(t=j+k);if(t, binomial(t,j)*(2^t + j*(2^t-1)/t), 1) ))} \\ _Andrew Howroyd_, Jan 20 2024

%Y Cf. A181132, A000788.

%K nonn,base

%O 1,2

%A _Tamas Sandor Nagy_, Jan 20 2024

%E More terms from _Andrew Howroyd_, Jan 20 2024