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A029931
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If 2n = Sum 2^e_i, a(n) = Sum e_i.
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308
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0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 11, 12, 13, 14, 14, 15, 16, 17, 15, 16, 17, 18, 18, 19, 20, 21, 7, 8, 9, 10, 10, 11, 12, 13, 11, 12, 13, 14, 14, 15, 16
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OFFSET
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0,3
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COMMENTS
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Write n in base 2, n = sum b(i)*2^(i-1), then a(n) = sum b(i)*i. - Benoit Cloitre, Jun 09 2002
May be regarded as a triangular array read by rows, giving weighted sum of compositions in standard order. The standard order of compositions is given by A066099. - Franklin T. Adams-Watters, Nov 06 2006
Sum of all positive integer roots m_i of polynomial {m,k} - see link [Shevelev]; see also A264613. - Vladimir Shevelev, Dec 13 2015
Also the sum of binary indices of n, where a binary index of n (A048793) is any position of a 1 in its reversed binary expansion. For example, the binary indices of 11 are {1,2,4}, so a(11) = 7. - Gus Wiseman, May 22 2024
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LINKS
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FORMULA
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a(n) = a(n - 2^L(n)) + L(n) + 1 [where L(n) = floor(log_2(n)) = A000523(n)] = sum of digits of A048794 [at least for n < 512]. - Henry Bottomley, Mar 09 2001
a(0) = 0, a(2n) = a(n) + e1(n), a(2n+1) = a(2n) + 1, where e1(n) = A000120(n). a(n) = log_2(A029930(n)). - Ralf Stephan, Jun 19 2003
G.f.: (1/(1-x)) * Sum_{k>=0} (k+1)*x^2^k/(1+x^2^k). - Ralf Stephan, Jun 23 2003
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EXAMPLE
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14 = 8+4+2 so a(7) = 3+2+1 = 6.
Composition number 11 is 2,1,1; 1*2+2*1+3*1 = 7, so a(11) = 7.
The triangle starts:
0
1
2 3
3 4 5 6
The reversed binary expansion of 18 is (0,1,0,0,1) with 1's at positions {2,5}, so a(18) = 2 + 5 = 7. - Gus Wiseman, Jul 22 2019
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MAPLE
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HammingWeight := n -> add(i, i = convert(n, base, 2)):
a := proc(n) option remember; `if`(n = 0, 0,
ifelse(n::even, a(n/2) + HammingWeight(n/2), a(n-1) + 1)) end:
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MATHEMATICA
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a[n_] := (b = IntegerDigits[n, 2]).Reverse @ Range[Length @ b]; Array[a, 78, 0] (* Jean-François Alcover, Apr 28 2011, after B. Cloitre *)
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PROG
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(PARI) for(n=0, 100, l=length(binary(n)); print1(sum(i=1, l, component(binary(n), i)*(l-i+1)), ", "))
(Haskell)
a029931 = sum . zipWith (*) [1..] . a030308_row
(Python)
def A029931(n): return sum(i if j == '1' else 0 for i, j in enumerate(bin(n)[:1:-1], 1)) # Chai Wah Wu, Dec 20 2022
(C#)
ulong result = 0, counter = 1;
while(n > 0) {
if (n % 2 == 1)
result += counter;
counter++;
n /= 2;
}
return result;
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CROSSREFS
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Contains exactly A000009(n) copies of n.
For product instead of sum we have A096111.
Numbers k such that a(k) is prime are A372689.
A014499 lists binary indices of prime numbers.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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