A305669 - OEIS

%I #11 Jun 28 2018 21:44:11

%S 0,1,2,3,4,5,6,7,8,9,11,13,14,17,21,32,41,51,53,54,65,81,85,95,98,101,

%T 108,109,116,171,179,210,321,632,811,910,917,1013,1071,1112,1113,1114,

%U 1116,1271,1291,1312,1313,1315,1316,1323,1375,1381,1415,1516,1517,1585

%N Take the sum of the digits of a number, put it at the left side and delete the same number of digits at the right side. Repeat the process. Sequence lists numbers that reach themselves after some steps.

%C Fixed points of the process (numbers that reach themselves in a single step) are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1818, 17171, 272727, 1313131, 2626262, 3939393, 12121212, 24242424, 36363636, 48484848, etc.

%C The number of steps to return to the original number or to enter another cycle depends on the number of digits. Here below all possible steps against the number of digits from 1 to 8:

%C Digits    Steps

%C 1         1

%C 2         3, 12

%C 3         3, 9

%C 4         1, 3, 10, 31

%C 5         1, 9

%C 6         1, 4, 13, 21, 39

%C 7         1, 2, 4, 5, 6, 10, 12, 20, 23, 30, 60

%C 8         1, 3, 26, 78

%p P:=proc(q) local a,b,c,d,k,n,x;

%p for n from 1 to q do a:=convert(n,base,10); d:=a; x:=0;

%p while x<10^(ilog10(n)+1) do x:=x+1; b:=convert(a,`+`);

%p c:=ilog10(b)+1; b:=convert(b,base,10);

%p for k from 1 to nops(a)-c do a[k]:=a[k+c]; od;

%p for k from 1 to c do a[nops(a)-c+k]:=b[k]; od;

%p if a=d then print(n); break; fi; od; od; end: P(10^6);

%Y Cf. A007953.

%K nonn,easy,base

%O 0,3

%A _Paolo P. Lava_, Jun 19 2018