A289868 Consider the digit reverse of a number x. Take the sum of these digits and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to x.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 21, 25, 42, 63, 84, 143, 286, 2355, 5821, 6618, 11709, 12482, 33747, 39571, 129109, 466957, 1162248, 1565166, 1968084, 3636638, 3853951, 4898376, 13443745, 13933175, 17118698, 22421197, 24153462, 147440984, 209989875, 245742153

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)

Links

Table of n, a(n) for n=0..40.

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

Cf. A000045, A000073, A001590, A004086, A248134, A269309, A269307, A269311.

Sequence in context: A372281 A124359 A124366 * A304247 A304245 A304244

Adjacent sequences: A289865 A289866 A289867 * A289869 A289870 A289871

Keyword

nonn,base

Author

Paolo P. Lava, Jul 14 2017

Status

approved