A308105 Numbers m such that m is greater than the sum of the k-th powers of its digits, where k is the number of digits of m.

10, 11, 12, 13, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 80, 81, 82, 83, 84, 90, 91, 92, 93, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134

Offset: 1

Author

Bernard Schott, May 13 2019

Keywords

• nonn
• base

Comments

These integers are called "nombres résistants" on the French site Diophante.

There exists a smallest number M_0 such that every number >= M_0 is a term of this sequence. This integer has 60 digits: M_0 = 102 * 10^57. So 102 * 10^57 - 1 is not "résistant" (proof in the link).

Examples

			34 - (3^2 + 4^2) = 9 so 34 is a term.
126 - (1^3 + 2^3 + 6^3) = -99 and 126 is not a term.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Diophante, A367. Les entiers font de la résistance, Oct. 2017 (in French).

Crossrefs

Cf. A000027, A101337, A005188.

Programs

Magma

sol:=[];v:=[];digit:=[]; m:=1;
for u in [1..150] do
        digit:=Intseq(u);
             for i in [1..#digit] do v[i]:=digit[i]^#digit; end for;
             if u-&+v gt 0 then sol[m]:=u; m:=m+1; end if;
end for;
sol; // Marius A. Burtea, May 13 2019

Maple

filter:= proc(n) local L,m,t;
  L:= convert(n,base,10);
  m:= nops(L);
  n > add(t^m,t=L)
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 21 2019

Mathematica

Select[Range[140], # - Total[IntegerDigits[#]^IntegerLength[#]] > 0 &] (* Michael De Vlieger, Jun 09 2019 *)

PARI

isok(n) = { my(d=digits(n), nb=#d); n > sum(k=1, #d, d[k]^nb);} \\ Michel Marcus, May 19 2019

Formula

Numbers m such that m - A101337(m) > 0.