A010061 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.

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

Offset: 1

Author

Leonid Broukhis

Keywords

• nonn
• base

Comments

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

References

• 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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• 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.
• Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
• Griffin N. Macris, Proof that no consecutive self numbers exist, 2020.
• G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers. II, Bollettino dell'Unione Matematica Italiana, 2-A (1999), 397-399.
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Complement of A228082, or equally, numbers which do not occur in A092391. Gives the positions of zeros (those occurring after a(0)) in A228085-A228087 and positions of ones in A227643. Leftmost column of A228083. Base-10 analog: A003052.

Cf. A010062, A055938, A230091, A230092, A230058, A242403.

Cf. A228088, A227915, A232228.

Programs

Haskell

a010061 n = a010061_list !! (n-1)
a010061_list = filter ((== 0) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

Maple

# For Maple code see A230091. - N. J. A. Sloane, Oct 10 2013

Mathematica

Table[n + Total[IntegerDigits[n, 2]], {n, 0, 300}] // Complement[Range[Last[#]], #]& (* Jean-François Alcover, Sep 03 2013 *)

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)) ) );
}

Extensions

More terms from Antti Karttunen, Aug 17 2013

Better definition from Matthew C. Russell, Oct 08 2013