OFFSET
0,3
COMMENTS
Numbers of iterations for the listed terms are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 5, 4, 5, 5, 5, 6, 6, 9, 10, 10, 11, 11, 12, 12, 14, 15, 17, 17, 17, 18, 18, 18, 20, 20, 20, 21, 21, 23, 23, 24.
From David A. Corneth, Jul 20 2017: (Start)
If n is in the sequence and its highest digit is m then n * (10\m) is in the sequence for m * (10\m) < 10.
Let T(q, k, n) = b(n) from the following recursion: for 0 <= i <= q-1, b(i) = 1 if i = k, else, b(i) = 0. Then b(n) = Sum(j=1..n, b(n-j)). If some m has q digits d1,..,dk,..,dq with d1 nonzero then after n iterations, we have Sum(j=1..q, T(q, k, n)*d(q-k+1)). This enables an iterative approach to finding solutions with q digits. (End)
EXAMPLE
Digit reverse of 17 is 71 and 7 + 1 = 8, 1 + 8 = 9, 8 + 9 = 17;
Digit reverse of 286 is 682 and 6 + 8 + 2 = 16, 8 + 2 + 16 = 26, 2 + 16 + 26 = 44, 16 + 26 + 44 = 86, 26 + 44 + 86 = 156, 44 + 86 + 156 = 286.
MAPLE
P:=proc(q) local a, b, k, n; for n from 0 to q do a:=convert(n, base, 10); b:=convert(a, `+`); while b<n do for k from 1 to nops(a)-1 do a[k]:=a[k+1]; od; a[nops(a)]:=b;
b:=convert(a, `+`); od; if b=n then print(n); fi; od; end: P(10^9);
MATHEMATICA
Select[Range[10^6], Function[n, Total@ NestWhile[Append[Drop[#, 1], Total@ #] &, Reverse@ IntegerDigits@ n, Total@ # < n &] == n]] (* Michael De Vlieger, Jul 20 2017 *)
PROG
(PARI) is(n) = {my(d = Vecrev(digits(n))); while(vecsum(d)<n, d = concat(vector(#d-1, i, d[i+1]), [vecsum(d)])); vecsum(d)==n} \\ David A. Corneth, Jul 20 2017
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jul 14 2017
STATUS
approved