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%I #16 May 30 2019 09:00:21
%S 10,20,30,40,50,60,70,80,83,90,93,103,113,123,133,143,163,166,176,180,
%T 186,196,206,249,253,259,269,270,326,332,343,360,416,433,450,489,506,
%U 523,540,579,596,613,630,652,669,686,703,720,730,732,742,747,752,762,772
%N Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)*d_i} = Sum_{i=1..j-1}{(j-i)*d_i}. Case x = 9.
%C All the palindromic numbers in base 9 with an even number of digits belong to the sequence.
%C Here the fulcrum is between two digits while in the sequence from A282107 to A282115 is one of the digits.
%C The first number with this property in all the bases from 2 to 9 is 1030854453981. - _Giovanni Resta_, Feb 16 2017
%H Paolo P. Lava, <a href="/A282150/b282150.txt">Table of n, a(n) for n = 1..10000</a>
%e 762 in base 9 is 1036. If we split the number in 103 and 6 we have 3*1 + 0*2 + 1*3 = 6 for the left side and 6*1 = 6 for the right one.
%p P:=proc(n,h) local a,j,k: a:=convert(n, base, h):
%p for k from 1 to nops(a)-1 do
%p if add(a[j]*(k-j+1),j=1..k)=add(a[j]*(j-k),j=k+1..nops(a))
%p then RETURN(n); break: fi: od: end: seq(P(i,9),i=1..10^3);
%Y Cf. A282107-A282115, A282143-A282149, A282151.
%K nonn,base,easy
%O 1,1
%A _Paolo P. Lava_, Feb 15 2017
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