A171798 a(n) = base-10 concatenation XYZ, where X = number of bits in binary expansion of n, Y = number of 0's, Z = number of 1's.

101, 211, 202, 321, 312, 312, 303, 431, 422, 422, 413, 422, 413, 413, 404, 541, 532, 532, 523, 532, 523, 523, 514, 532, 523, 523, 514, 523, 514, 514, 505, 651, 642, 642, 633, 642, 633, 633, 624, 642, 633, 633, 624, 633, 624, 624, 615, 642, 633, 633, 624

Offset

1,1

Comments

Start with n, repeatedly apply the map i -> a(i). Then every n converges to one of 1019, 1147, 1165 or 14311 (cf. A171813). Proof: this is true by direct calculation for n=1..2^14. For larger n, a(n) < n.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Example

14 = 1110 in base 2, so X=4, Y=1, Z=3, a(14)=413.

Maple

# Maple code for trajectories of numbers from 1 to M:
F:=proc(n) local s, t1, t2; t1:=convert(n, base, 2); t2:=nops(t1); s:=add(t1[i], i=1..t2);
parse(cat(t2, t2-s, s)); end;
M:=16384;
for n from 1 to M do t3:=F(n); sw:=-1;
for i from 1 to 10 do
if (t3 = 1147) or (t3 = 1165) or (t3 = 1019) or (t3 = 14311) then sw:=1; break; fi;
t3:=F(t3);
od;
if sw < 0 then lprint(n); fi;
od:
Contribution from R. J. Mathar, Oct 15 2010: (Start)
read("transforms") ; cat2 := proc(a, b) dgsb := max(1, ilog10(b)+1) ; a*10^dgsb+b ; end proc:
catL := proc(L) local a; a := op(1, L) ; for i from 2 to nops(L) do a := cat2(a, op(i, L)) ; end do; a; end proc:
A070939 := proc(n) max(1, ilog2(n)+1) ; end proc:
A171798 := proc(n) local n1, n3 ; n1 := A070939(n) ; n3 := wt(n) ; catL([n1, n1-n3, n3]) ; end proc:
seq(A171798(n), n=1..80) ; (End)

Mathematica

ans[n_]:=Module[{idn2=IntegerDigits[n, 2]}, FromDigits[{Length[idn2], Count[idn2, 0], Count[idn2, 1]}]]; Table[ans[i], {i, 50}] (* Harvey P. Dale, Nov 06 2010 *)

Prog

(Haskell)
a171798 n = read $ concatMap (show . ($ n))
                   [a070939, a023416, a000120] :: Integer
-- Reinhard Zumkeller, Feb 22 2012
(Python)
def a(n):
    b = bin(n)[2:]
    z = b.count("0")
    return int(str(len(b)) + str(z) + str(len(b)-z))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Mar 28 2022

Crossrefs

Cf. A171797, A171813, A171823, A171825.

Cf. A070939, A023416, A000120.

Sequence in context: A357033 A319744 A303575 * A067861 A142633 A142838

Adjacent sequences: A171795 A171796 A171797 * A171799 A171800 A171801

Keyword

nonn,base,easy

Author

N. J. A. Sloane, Oct 15 2010, Oct 16 2010

Extensions

More terms from R. J. Mathar, Oct 15 2010

Status

approved