A171823 -id:A171823 - cp's OEIS frontend

A171825 Trajectory of 1 under repeated application of i -> A171823(i).

1, 101, 11110, 1011100, 11111100, 100010110, 1001101100, 1010101101, 1010100110, 1010101101, 1010100110, 1010101101, 1010100110, 1010101101, 1010100110, 1010101101, 1010100110, 1010101101, 1010100110, 1010101101

Offset: 1

N. J. A. Sloane, Oct 16 2010

Has become periodic with period {1010101101,1010100110}.

# Apply map to base-10 number
F:=proc(n) local t1,t2,t2b,n1,n1b,n0,n0b,t3,t4;
t1:=convert(n,base,2);
t2:=nops(t1);
t2b:=convert(t2,base,2);
n1:=add(t1[i],i=1..t2);
n1b:=convert(n1,base,2);
n0:=t2-n1;
n0b:=convert(n0,base,2);
t3:=[
seq(t2b[nops(t2b)+1-i],i=1..nops(t2b)),
seq(n0b[nops(n0b)+1-i],i=1..nops(n0b)),
seq(n1b[nops(n1b)+1-i],i=1..nops(n1b))
];
t4:="";
for i from 1 to nops(t3) do t4:=cat(t4,t3[i]); od:
parse(t4);
end;
# Apply map to binary number
G:=proc(n) local t2,t2b,n1,n1b,n0,n0b,t3,t4; global digsum;
t2:=length(n);
t2b:=convert(t2,base,2);
n1:=digsum(n);
n1b:=convert(n1,base,2);
n0:=t2-n1;
n0b:=convert(n0,base,2);
t3:=[
seq(t2b[nops(t2b)+1-i],i=1..nops(t2b)),
seq(n0b[nops(n0b)+1-i],i=1..nops(n0b)),
seq(n1b[nops(n1b)+1-i],i=1..nops(n1b))
];
t4:="";
for i from 1 to nops(t3) do t4:=cat(t4,t3[i]); od:
parse(t4);
end;
# Then do 1, F(1), G(F(1)), G(%), G(%), ... .

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.

Original entry on oeis.org

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

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

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.

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

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

Cf. A171797, A171813, A171823, A171825.

Cf. A070939, A023416, A000120.

a171798 n = read $ concatMap (show . ($ n))
                   [a070939, a023416, a000120] :: Integer
-- Reinhard Zumkeller, Feb 22 2012

# 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)

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 *)

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

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

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A171825 Trajectory of 1 under repeated application of i -> A171823(i).

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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.

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