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

%I A330072 #42 May 19 2025 23:37:58
%S A330072 0,1,3,5,7,10,13,16,15,19,24,28,29,33,38,42,31,36,43,48,52,57,64,69,
%T A330072 61,66,73,78,82,87,94,99,63,69,78,84,91,97,106,112,108,114,123,129,
%U A330072 136,142,151,157,125,131,140,146,153,159,168,174,170,176,185,191,198
%N A330072 a(n) is the sum of all integers whose binary representation is contained in the binary representation of n (with multiplicity).
%F A330072 a(2^k) = 2^(k+1)-1.
%F A330072 a(2^k-1) = (8*2^k-8-5*k-k^2)/2. - _Giovanni Resta_, Dec 02 2019
%F A330072 From _David Radcliffe_, May 14 2025: (Start)
%F A330072 a(n) = Sum_{i=0..b} Sum_{j=0..b-i} ([n/2^i] mod 2^j) where b is the bit length of n.
%F A330072 a(n) - 3a([n/2]) + 2a([n/4]) = A070939(n) if n is odd, 0 otherwise. (End)
%e A330072 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.
%t A330072 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 *)
%o A330072 (Python)
%o A330072 def bitlist(n):
%o A330072     output = []
%o A330072     while n != 0:
%o A330072         output.append(n % 2)
%o A330072         n //= 2
%o A330072     return output
%o A330072 #converts a number into a list of the digits in binary reversed
%o A330072 def bitsum(bitlist):
%o A330072     output = 0
%o A330072     for bit in bitlist:
%o A330072         if bit == 1:
%o A330072             output += 1
%o A330072     return output
%o A330072 #gives the cross sum of a bitlist
%o A330072 def a(bitlist):
%o A330072     output = 0
%o A330072     l = len(bitlist)
%o A330072     for x in range(l):
%o A330072         output += bitlist[x] * (l - x) * (2**(x + 1) - 1)
%o A330072     return output
%o A330072 #to get the first 60 numbers of the sequence, write:
%o A330072 for x in range(0, 60):
%o A330072     print(a(bitlist(x)))
%o A330072 (Python)
%o A330072 def a(n):
%o A330072     b = n.bit_length()
%o A330072     return sum((n//2**i) % (2**j) for i in range(b+1) for j in range(b-i+1))
%o A330072 # _David Radcliffe_, May 14 2025
%Y A330072 Cf. A007088, A005187, A049802, A078823, A225580 (base-10 version).
%K A330072 nonn,base
%O A330072 0,3
%A A330072 _Tonio Kettner_, Nov 30 2019