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A010061
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Binary self or Colombian numbers: numbers that cannot be expressed as the sum of distinct terms of the form 2^k+1 (k>=0), or equivalently, numbers not of form m + sum of binary digits of m.
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43
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1, 4, 6, 13, 15, 18, 21, 23, 30, 32, 37, 39, 46, 48, 51, 54, 56, 63, 71, 78, 80, 83, 86, 88, 95, 97, 102, 104, 111, 113, 116, 119, 121, 128, 130, 133, 135, 142, 144, 147, 150, 152, 159, 161, 166, 168, 175, 177, 180, 183, 185, 192, 200, 207, 209, 212, 215, 217
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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No two consecutive values appear in this sequence (see Links). - Griffin N. Macris, May 31 2020
The asymptotic density of this sequence is (1/8) * (2 - Sum_{n>=1} 1/2^a(n))^2 = 0.252660... (A242403). - Amiram Eldar, Nov 28 2020
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24, pp. 179-180.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.
G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers, Bolletino U. M. I. (7) 9-A (1995), 143-148.
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LINKS
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Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365 [math.NT], 2021-2022; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
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MAPLE
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MATHEMATICA
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Table[n + Total[IntegerDigits[n, 2]], {n, 0, 300}] // Complement[Range[Last[#]], #]& (* Jean-François Alcover, Sep 03 2013 *)
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PROG
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(Haskell)
a010061 n = a010061_list !! (n-1)
a010061_list = filter ((== 0) . a228085) [1..]
/* PARI: Gen(n, b) returns a list of the generators of n in base b. Written by Max Alekseyev (see Alekseyev et al., 2021).
For example, Gen(101, 10) returns [91, 101]. - N. J. A. Sloane, Jan 02 2022 */
{ Gen(u, b=10) = my(d, m, k);
if(u<0 || u==1, return([]); );
if(u==0, return([0]); );
d = #digits(u, b)-1;
m = u\b^d;
while( sumdigits(m, b) > u - m*b^d,
m--;
if(m==0, m=b-1; d--; );
);
k = u - m*b^d - sumdigits(m, b);
vecsort( concat( apply(x->x+m*b^d, Gen(k, b)),
apply(x->m*b^d-1-x, Gen((b-1)*d-k-2, b)) ) );
}
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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