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A283531 Number of steps to return to n through a chain-addition sequence mod 10 with window of size equal to the number of digits of n.

%I A283531 #11 Mar 14 2017 06:23:26
%S A283531 1,1,1,1,1,1,1,1,1,1,60,60,60,12,60,60,60,60,12,60,20,12,20,60,20,60,
%T A283531 4,60,20,60,60,60,60,60,12,60,60,60,60,12,20,60,4,60,20,60,20,12,20,
%U A283531 60,3,60,60,60,60,3,60,60,60,60,20,60,20,12,20,60,20,60,4,60
%N A283531 Number of steps to return to n through a chain-addition sequence mod 10 with window of size equal to the number of digits of n.
%C A283531 Alternative definition: number of steps to return to n under a transform where the MSD(n) is deleted and as LSD(n) is concatenated Sd(n) mod 10, where Sd(n) is the sum of the digits of n.
%C A283531 Numbers n that need exactly n steps are 1, 20, 124, 1560.
%C A283531 Number of steps to return to 10^k, with k = 0, 1, 2, ..., are listed in A181190.
%H A283531 Paolo P. Lava, <a href="/A283531/b283531.txt">Table of n, a(n) for n = 0..10000</a>
%e A283531 a(18) = 12 because:
%e A283531 (1 + 8) mod 10 = 9 -> 89;
%e A283531 (8 + 9) mod 10 = 7 -> 97;
%e A283531 (9 + 7) mod 10 = 6 -> 76;
%e A283531 (7 + 6) mod 10 = 3 -> 63;
%e A283531 (6 + 3) mod 10 = 9 -> 39;
%e A283531 (3 + 9) mod 10 = 2 -> 92;
%e A283531 (9 + 2) mod 10 = 1 -> 21;
%e A283531 (2 + 1) mod 10 = 3 -> 13;
%e A283531 (1 + 3) mod 10 = 4 -> 34;
%e A283531 (3 + 4) mod 10 = 7 -> 47;
%e A283531 (4 + 7) mod 10 = 1 -> 71;
%e A283531 (7 + 1) mod 10 = 8 -> 18;
%e A283531 a(68) = 4 because:
%e A283531 (6 + 8) mod 10 = 4 -> 84;
%e A283531 (8 + 4) mod 10 = 2 -> 42;
%e A283531 (4 + 2) mod 10 = 6 -> 26;
%e A283531 (2 + 6) mod 10 = 8 -> 68.
%p A283531 S:=proc(w) local x, y, z; x:=w; y:=0; for z from 1 to ilog10(x)+1 do y:=y+(x mod 10); x:=trunc(x/10); od; y mod 10; end: P:=proc(q) local a,k,n; for n from 0 to q do a:=n;
%p A283531 for k from 1 to q do a:=10*(a mod 10^(ilog10(n)))+S(a); if a=n then print(k);
%p A283531 break; fi; od; od; end: P(10^5);
%Y A283531 Cf. A007953, A181190.
%K A283531 nonn,base,easy
%O A283531 0,11
%A A283531 _Paolo P. Lava_, Mar 10 2017