A330072 a(n) is the sum of all integers whose binary representation is contained in the binary representation of n (with multiplicity).

0, 1, 3, 5, 7, 10, 13, 16, 15, 19, 24, 28, 29, 33, 38, 42, 31, 36, 43, 48, 52, 57, 64, 69, 61, 66, 73, 78, 82, 87, 94, 99, 63, 69, 78, 84, 91, 97, 106, 112, 108, 114, 123, 129, 136, 142, 151, 157, 125, 131, 140, 146, 153, 159, 168, 174, 170, 176, 185, 191, 198

Offset

0,3

Links

Table of n, a(n) for n=0..60.

Formula

a(2^k) = 2^(k+1)-1.

a(2^k-1) = (8*2^k-8-5*k-k^2)/2. - Giovanni Resta, Dec 02 2019

Example

For n = 5: 5 in binary is 101, so a(n) = 101 + 10 + 01 + 1 + 0 + 1 in binary, which is 5 + 2 + 1 + 1 + 0 + 1 = 10.

Mathematica

a[n_] := Block[{d = IntegerDigits[n, 2]}, Sum[FromDigits[Take[d, {i, j}], 2], {j, Length[d]}, {i, j}]]; Array[a, 61, 0] (* Giovanni Resta, Dec 02 2019 *)

Prog

(Python)
def bitlist(n):
    output = []
    while n != 0:
        output.append(n % 2)
        n /= 2
    return output
#converts a number into a list of the digits in binary reversed
def bitsum(bitlist):
    output = 0
    for bit in bitlist:
        if bit == 1:
            output += 1
    return output
#gives the cross sum of a bitlist
def a(bitlist):
    output = 0
    l = len(bitlist)
    for x in range(l):
        output += bitlist[x] * (l - x) * (2**(x + 1) - 1)
    return output
#to get the first n numbers of the sequence, write:
for x in range(1, n + 1):
    print x, a(bitlist(x))

Crossrefs

Cf. A007088, A078823, A225580.

Sequence in context: A152004 A298862 A198409 * A213510 A130257 A276498

Adjacent sequences: A330069 A330070 A330071 * A330073 A330074 A330075

Keyword

nonn,base

Author

Tonio Kettner, Nov 30 2019

Status

approved