A292513 A number N is called "docile" if there are two integers a and b such that N = a + b with a > b > 0 and S(a) = S(b) where S(n) is the sum of the digits of the number n.

11, 13, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 99, 101

Offset

1,1

Comments

A number which is not "docile" is called a "rebel". These definitions come from the French site Diophante, see link.

There are an infinite number of odd docile numbers and also and infinite number of even docile numbers. For instance 10^n + 1 = 1000...00001 and S(10^n) = S(1) = 1, or, 2000...00002 = 2000...0000 + 2 and S(2000..000)= S(2) = 2.

What are the smallest integers which are 2 times, 3 times, ... docile numbers?

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maurice Bauval, Dociles et Rebelles, Diophante, A 348, September 2014

Example

15 is docile because 15 = 12 + 3 and S(12) = S(3) = 3.
16 is not docile because 16 = 15 + 1 = 14 + 2 = 13 + 3 = 12 + 4 = 11 + 5 = 10 + 6 = 9 + 7 and never S(a) = S(b) with these integers.

Maple

N:= 200: # for all terms <= N
A:= {}:
for x from 1 to N do
  t:= convert(convert(x, base, 10), `+`);
  if not assigned(S[t]) then S[t]:= {} fi;
  A:= A union select(`<=`, map(`+`, S[t], x), N);
  S[t]:= S[t] union {x};
od:
sort(convert(A, list)); # Robert Israel, Sep 23 2019

Mathematica

Select[Range@ 101, Count[IntegerPartitions[#, {2}], _?(And[#1 > #2, Total@ IntegerDigits@ #1 == Total@ IntegerDigits@ #2] & @@ # &)] > 0 &] (* Michael De Vlieger, Sep 18 2017 *)

Prog

(PARI) isok(n) = for (x=1, n\2, if ((x != (n-x)) && (sumdigits(x) == sumdigits(n-x)), return (1)); ); return (0); \\ Michel Marcus, Sep 18 2017

Crossrefs

Cf. A292514 ("rebel" numbers).

Sequence in context: A228709 A358270 A358076 * A171491 A277268 A337254

Adjacent sequences: A292510 A292511 A292512 * A292514 A292515 A292516

Keyword

nonn,base

Author

Bernard Schott, Sep 18 2017

Status

approved