A308105 - OEIS

%I #44 Sep 08 2022 08:46:21

%S 10,11,12,13,20,21,22,23,24,30,31,32,33,34,35,40,41,42,43,44,45,50,51,

%T 52,53,54,55,60,61,62,63,64,65,70,71,72,73,74,75,80,81,82,83,84,90,91,

%U 92,93,100,101,102,103,104,110,111,112,113,114,120,121,122,123,124,130,131,132,133,134

%N 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.

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

%C 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).

%H Robert Israel, <a href="/A308105/b308105.txt">Table of n, a(n) for n = 1..10000</a>

%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a3-nombres-remarquables/3969-a367-les-entiers-font-de-la-resistance">A367. Les entiers font de la résistance</a>, Oct. 2017 (in French).

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

%e 34 - (3^2 + 4^2) = 9 so 34 is a term.

%e 126 - (1^3 + 2^3 + 6^3) = -99 and 126 is not a term.

%p filter:= proc(n) local L,m,t;

%p   L:= convert(n,base,10);

%p   m:= nops(L);

%p   n > add(t^m,t=L)

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Jun 21 2019

%t Select[Range[140], # - Total[IntegerDigits[#]^IntegerLength[#]] > 0 &] (* _Michael De Vlieger_, Jun 09 2019 *)

%o (Magma) sol:=[];v:=[];digit:=[]; m:=1;

%o for u in [1..150] do

%o         digit:=Intseq(u);

%o              for i in [1..#digit] do v[i]:=digit[i]^#digit; end for;

%o              if u-&+v gt 0 then sol[m]:=u; m:=m+1; end if;

%o end for;

%o sol; // _Marius A. Burtea_, May 13 2019

%o (PARI) isok(n) = { my(d=digits(n), nb=#d); n > sum(k=1, #d, d[k]^nb);} \\ _Michel Marcus_, May 19 2019

%Y Cf. A000027, A101337, A005188.

%K nonn,base

%O 1,1

%A _Bernard Schott_, May 13 2019