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Numbers x such that x = Sum_{i=1..k} (x mod d_(x-i)) for some k, where d_(x-i) is the aliquot parts of (x-i).
3

%I #10 Aug 07 2017 08:54:02

%S 11,13,14,15,18,40,60,83,205,226,234,244,267,310,321,336,341,462,543,

%T 572,610,757,766,771,802,826,919,968,993,1089,1366,1391,1734,1758,

%U 1863,1911,1985,1993,2095,2222,2396,2405,2422,2522,3495,3634,3655,3672,3823,3870

%N Numbers x such that x = Sum_{i=1..k} (x mod d_(x-i)) for some k, where d_(x-i) is the aliquot parts of (x-i).

%C Values of k for the listed terms are 5, 7, 6, 9, 10, 7, 8, 7, 11, 11, 12, 12, 12, 13, 13, 15, 14, 17, 15, 18, 16, 20, 18, 19, 20, 20, 19, 22, 21, 23, 24, 25, 26, 29, 28, 28, 29, 30, 29, 30, 31, 29, 30, 33, 37, 36, 39, 39, 41, 41, ...

%H Paolo P. Lava, <a href="/A290468/b290468.txt">Table of n, a(n) for n = 1..500</a>

%e For 11 the value of k is 5. Aliquot parts of 10, 9, 8, 7 and 6 are: [1, 2, 5], [1, 3], [1, 2, 4], [1], [1, 2, 3]. Residues are 0 + 1 + 1 + 0 + 2 + 0 + 1 + 3 + 0 + 0 + 1 + 2 that sum up to 11.

%p with(numtheory): P:=proc(q) local a, b, j, k, n; for n from 6 to q do

%p a:=0; k:=0; while a<n do k:=k+1; b:=sort([op(divisors(n-k))]);

%p a:=a+add(n mod b[j], j=1..nops(b)-1); od;

%p if a=n then print(n); fi; od; end: P(10^9);

%Y Cf. A286873, A290469.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Aug 03 2017