A292514 A number N is called a "rebel" number if there do not exist 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.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 29, 38, 40, 49, 58, 60, 69, 78, 80, 89, 98, 100, 199, 399, 599, 799, 999, 2999, 4999, 6999, 8999, 19999, 39999, 59999, 79999, 99999, 299999, 499999, 699999, 899999, 1999999, 3999999, 5999999, 7999999, 9999999

Offset

1,2

Comments

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

There are an infinite number of odd "rebel" numbers. For instance, all the repdigits 9999999...99 with a string of (2k+1) times the digit 9; but there are only eighteen even "rebel" numbers: {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 38, 40, 58, 60, 78, 80, 98, 100}.

Links

Table of n, a(n) for n=1..50.

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

Example

14 is rebel because 14 = 13 + 1 = 12 + 2 = 11 + 3 = 10 + 4 = 9 + 5 = 8 + 6 and never S(a) = S(b) with these integers.

Mathematica

Select[Range@ 100, Count[IntegerPartitions[#, {2}], _?(And[#1 > #2, Total@ IntegerDigits@ #1 == Total@ IntegerDigits@ #2] & @@ # &)] == 0 &]~Join~Union@ Flatten@ Table[Map[FromDigits[{#}~Join~ConstantArray[9, k]] &, Range[1, 9, 2] - Boole[OddQ@ k]], {k, 2, 6}] (* 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 (0)); ); return (1); } \\ Michel Marcus, Sep 18 2017

Crossrefs

Cf. A292513 ("docile" numbers).

Sequence in context: A130232 A103969 A271168 * A030141 A242367 A355596

Adjacent sequences: A292511 A292512 A292513 * A292515 A292516 A292517

Keyword

nonn,base

Author

Bernard Schott, Sep 18 2017

Extensions

More terms from Giovanni Resta, Sep 18 2017

Status

approved