A110382 Numbers which are sum of distinct unary numbers (containing only ones), i.e., numbers which are sum of distinct numbers of the form (10^k - 1)/9.

1, 11, 12, 111, 112, 122, 123, 1111, 1112, 1122, 1123, 1222, 1223, 1233, 1234, 11111, 11112, 11122, 11123, 11222, 11223, 11233, 11234, 12222, 12223, 12233, 12234, 12333, 12334, 12344, 12345, 111111, 111112, 111122, 111123, 111222, 111223

Offset: 1

Author

Amarnath Murthy, Jul 25 2005

Keywords

• easy
• nonn

Comments

Not the same as A096299, since a(1023) = 1234567900 which is not in lexicographic order. - Ralf Stephan, May 17 2007

Links

• Georg Fischer, Table of n, a(n) for n = 1..16384 [First 1023 terms from David A. Corneth]

Crossrefs

Cf. A096299.

Programs

Maple

f:= proc(n) local L,i:
  L:= convert(n,base,2);
  add(L[i]*(10^i-1)/9, i=1..nops(L))
end proc:
map(f, [$1..100]); # Robert Israel, Feb 03 2025

Mathematica

Nest[Append[#1, 10 #1[[Floor[#2/2] ]] + DigitCount[#2, 2, 1]] & @@ {#, Length[#] + 1} &, {1}, 36] (* Michael De Vlieger, Mar 12 2021 *)

PARI

a(n) = sum(k=0, log(n)\log(2), hammingweight(n\(2^k))*10^k); \\ Michel Marcus, May 09 2019

PARI

a(n) = my(b = Vecrev(binary(n))); sum(i = 1, #b, b[i] * (10^i-1)) / 9 \\ David A. Corneth, May 19 2019

Formula

G.f.: 1/(1-x) * Sum_{k>=0} (10^(k+1) - 1)/9 * x^2^k/(1 + x^2^k). - Ralf Stephan, May 17 2007

a(n) = 10*a(floor(n/2)) + A000120(n) = Sum_{k=0..floor(log_2(n))} A000120(floor(n/(2^k)))*10^k. - Mikhail Kurkov, May 08 2019

a(n) = a(floor(n/2)) + A007088(n) = (10*A007088(n) - A000120(n))/9. - Mikhail Kurkov, Mar 03 2021

Extensions

a(1024) ff. corrected by Georg Fischer, Feb 03 2025