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with(numtheory): P:=proc(q, n, h) local a, b, d, j, k, : a:=convert(n, s; base, h):

for n k from 1 to q nops(a)-1 do a:=convert(n, base, h);

for k from 1 to trunc(nops(a)/2) do b:=a[k]; a[k]:=a[nops(a)-k+1]; a[nops(a)-k+1]:=b; od;

for k from 1 to nops(a)-1 do d:=0; s:=0;

for j from 1 to k do if a[j]>0 then s:=s+a[j]*(k-j+1); fi; od; for j from nops(a) by -1 to k+1 do

if a[j]>0 then d:=d+a[j]*(j-k); fi; od; if d=s then print(n); break; fi; od; od; end: P(10^3, 2);

if add(a[j]*(k-j+1), j=1..k)=add(a[j]*(j-k), j=k+1..nops(a))

then RETURN(n); break: fi: od: end: seq(P(i, 2), i=1..10^3);

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proposed

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proposed

Paolo P. Lava, _Giovanni Resta_, Feb 07 2017

allocated for Paolo P. Lava

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 = 2.

3, 6, 9, 12, 15, 18, 19, 24, 25, 30, 33, 36, 38, 45, 48, 50, 51, 60, 63, 66, 69, 72, 75, 76, 81, 87, 90, 96, 100, 102, 105, 117, 120, 126, 129, 131, 132, 138, 143, 144, 150, 152, 153, 162, 165, 174, 179, 180, 189, 192, 193, 195, 200, 204, 205, 210, 219, 231, 234

1,1

All the palindromic numbers in base 2 with an even number of digits belong to the sequence.

Here the fulcrum is between two digits while in the sequence from A282107 to A282115 is one of the digits.

Paolo P. Lava, <a href="/A282143/b282143.txt">Table of n, a(n) for n = 1..10000</a>

143 in base 2 is 10001111. If we split the number in 10001 and 111 we have 1*1 + 0*2 + 0*3 + 0*4 + 1*5 = 6 for the left side and 1*1 + 1*2 + 1*3 = 6 for the right one.

with(numtheory): P:=proc(q, h) local a, b, d, j, k, n, s;

for n from 1 to q do a:=convert(n, base, h);

for k from 1 to trunc(nops(a)/2) do b:=a[k]; a[k]:=a[nops(a)-k+1]; a[nops(a)-k+1]:=b; od;

for k from 1 to nops(a)-1 do d:=0; s:=0;

for j from 1 to k do if a[j]>0 then s:=s+a[j]*(k-j+1); fi; od; for j from nops(a) by -1 to k+1 do

if a[j]>0 then d:=d+a[j]*(j-k); fi; od; if d=s then print(n); break; fi; od; od; end: P(10^3, 2);

Cf. A282107 - A282115, A282144 - A282151.

allocated

nonn,base,easy

Paolo P. Lava, Feb 07 2017

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allocating

allocated

allocated for Paolo P. Lava

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