cp's OEIS Frontend

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.

%I A110382 #46 Feb 03 2025 13:55:50
%S A110382 1,11,12,111,112,122,123,1111,1112,1122,1123,1222,1223,1233,1234,
%T A110382 11111,11112,11122,11123,11222,11223,11233,11234,12222,12223,12233,
%U A110382 12234,12333,12334,12344,12345,111111,111112,111122,111123,111222,111223
%N 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.
%C A110382 Not the same as A096299, since a(1023) = 1234567900 which is not in lexicographic order. - _Ralf Stephan_, May 17 2007
%H A110382 Georg Fischer, <a href="/A110382/b110382.txt">Table of n, a(n) for n = 1..16384</a> [First 1023 terms from David A. Corneth]
%F A110382 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
%F A110382 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
%F A110382 a(n) = a(floor(n/2)) + A007088(n) = (10*A007088(n) - A000120(n))/9. - _Mikhail Kurkov_, Mar 03 2021
%p A110382 f:= proc(n) local L,i:
%p A110382   L:= convert(n,base,2);
%p A110382   add(L[i]*(10^i-1)/9, i=1..nops(L))
%p A110382 end proc:
%p A110382 map(f, [$1..100]); # _Robert Israel_, Feb 03 2025
%t A110382 Nest[Append[#1, 10 #1[[Floor[#2/2] ]] + DigitCount[#2, 2, 1]] & @@ {#, Length[#] + 1} &, {1}, 36] (* _Michael De Vlieger_, Mar 12 2021 *)
%o A110382 (PARI) a(n) = sum(k=0, log(n)\log(2), hammingweight(n\(2^k))*10^k); \\ _Michel Marcus_, May 09 2019
%o A110382 (PARI) a(n) = my(b = Vecrev(binary(n))); sum(i = 1, #b, b[i] * (10^i-1)) / 9 \\ _David A. Corneth_, May 19 2019
%Y A110382 Cf. A096299.
%K A110382 easy,nonn
%O A110382 1,2
%A A110382 _Amarnath Murthy_, Jul 25 2005
%E A110382 a(1024) ff. corrected by _Georg Fischer_, Feb 03 2025