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A228085 a(n) = number of distinct k which satisfy n = k + wt(k), where wt(k) (A000120) gives the number of 1's in binary representation of a nonnegative integer k.

%I A228085 #63 Sep 15 2024 22:07:44
%S A228085 1,0,1,1,0,2,0,1,1,1,1,1,1,0,2,0,1,2,0,2,1,0,2,0,1,1,1,1,1,1,0,2,0,2,
%T A228085 1,1,2,0,2,0,1,1,1,1,1,1,0,2,0,1,2,0,2,1,0,2,0,1,1,1,1,1,1,0,2,1,1,2,
%U A228085 1,1,2,0,1,1,1,1,1,1,0,2,0,1,2,0,2,1,0
%N A228085 a(n) = number of distinct k which satisfy n = k + wt(k), where wt(k) (A000120) gives the number of 1's in binary representation of a nonnegative integer k.
%C A228085 wt(k) is also called bitcount(k).
%C A228085 a(n) = number of times n occurs in A092391.
%C A228085 The first 2 occurs at n = A230303(2) = 5 (as we have two solutions A092391(3) = A092391(4) = 5).
%C A228085 The first 3 occurs at n = A230303(3) = 129 (as we have three solutions A092391(123) = A092391(124) = A092391(128) = 129).
%C A228085 The first 4 occurs at n = A230303(4) = 4102, where we have solutions A092391(4091) = A092391(4092) = A092391(4099) = A092391(4100) = 4102.
%C A228085 For n>=1, a(2^n) = a(n-1) since an integer k = m is a solution to n-1 = m + wt(m) if and only if k = 2^n - 1 - m is a solution to 2^n = k + wt(k). - _Max Alekseyev_, Feb 23 2021
%H A228085 Antti Karttunen, <a href="/A228085/b228085.txt">Table of n, a(n) for n = 0..8191</a>
%H A228085 Max A. Alekseyev and N. J. A. Sloane, <a href="https://arxiv.org/abs/2112.14365">On Kaprekar's Junction Numbers</a>, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
%H A228085 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%p A228085 For Maple code see A230091. - _N. J. A. Sloane_, Oct 10 2013
%p A228085 # Find all inverses of m under x -> x + wt(x) - _N. J. A. Sloane_, Oct 19 2013
%p A228085 A000120 := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120;
%p A228085 F:=proc(m) local ans,lb,n,i;
%p A228085 lb:=m-ceil(log(m+1)/log(2)); ans:=[];
%p A228085 for n from max(1,lb) to m do if (n+wt(n)) = m then ans:=[op(ans),n]; fi; od:
%p A228085 [seq(ans[i],i=1..nops(ans))];
%p A228085 end;
%t A228085 nmax = 8191; Clear[a]; a[_] = 0;
%t A228085 Scan[Set[a[#[[1]]], #[[2]]]&, Tally[Table[n + DigitCount[n, 2, 1], {n, 0, nmax}]]];
%t A228085 a /@ Range[0, nmax] (* _Jean-François Alcover_, Oct 29 2019 *)
%t A228085 a[n_] := Module[{k, cnt = 0}, For[k = n - Floor[Log[2, n]] - 1, k < n, k++, If[n == k + DigitCount[k, 2, 1], cnt++]]; cnt];
%t A228085 a /@ Range[0, 100] (* _Jean-François Alcover_, Nov 28 2020 *)
%o A228085 (Haskell)
%o A228085 a228085 n = length $ filter ((== n) . a092391) [n - a070939 n .. n]
%o A228085 -- _Reinhard Zumkeller_, Oct 13 2013
%Y A228085 A010061 gives the position of zeros, A228082 the positions of nonzeros, A228088 the positions of ones.
%Y A228085 Cf. A000120, A010062, A092391, A228086, A228087, A228091 (positions of 2's), A227643, A230058, A230092 (positions of 3's), A230093, A227915 (positions of 4's), A070939, A230303.
%K A228085 nonn,base
%O A228085 0,6
%A A228085 _Antti Karttunen_, Aug 09 2013