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Numbers x such that x = Sum_{i=1..k} (x mod d_(x-i)) + Sum_{i=1..k} (x mod d_(x+i)) for some k, where d_(x-i) and d_(x+i) are the aliquot parts of (x-i) and (x+i).
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%I #20 Aug 04 2017 15:41:25

%S 7,10,16,27,75,87,109,120,151,1887,4029,5829,17815,39780,62485,238021,

%T 254011,437744,779391,873565,979389,1713591,2409697,4194303,4199029,

%U 4607295,8353791,9928791,15370303,21381096,33653887,114203775,124540389,2146926591,6521655540

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

%C Values of k for the listed terms are 2, 2, 2, 2, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, ...

%C If d_(x-i) were the aliquot parts of (x-i) and d_(x+i) the divisors of (x+i) we would get the average of twin prime pairs (A014574).

%C If d_(x-i) were the divisors of (x-i) and d_(x+i) the aliquot parts of (x+i) we would get 5, 6, 7, 1296, 3228, 32767, 65784, 128766, 711236, ...

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

%p with(numtheory): P:=proc(q) local a,b,c,j,k,n;

%p for n from 3 to q do a:=0; k:=0; while a<n do k:=k+1;

%p b:=sort([op(divisors(n+k))]); c:=sort([op(divisors(n-k))]);

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

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

%Y Cf. A014574, A290468, A290469.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Aug 02 2017

%E a(27)-a(35) from _Giovanni Resta_, Aug 03 2017