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T:=proc(w) local x, y, z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a, b, n; for n from 1 to q do a:=n mod 10; b:=trunc(n/10^ilog10(n));
if (a=1 and b>1) or (a=6 and (b=2 or b=4 or b=6 or b=8)) or (b=5 and (a=3 or a=5 or a=7 or a=9)) then
if T(n)=b*(n mod 10^ilog10(n)) then print(n); fi; fi; od; end: P(10^10);
# alternative:
N:= 20: # to get all terms with at most N digits.
extend:= proc(d, psol, eqs)
local peqs, cvars, bvars, ncs, res, T, cs, ceqs, sol, svals;
peqs:= subs(psol, eqs);
cvars, bvars:= selectremove(t -> op(0, t) = 'c', indets(peqs));
ncs:= nops(cvars);
res:= NULL;
if ncs >= 1 then
T:= combinat:-cartprod([[$0..d-1]$ncs]);
while not T[finished] do
cs:= T[nextvalue]();
cs:= seq(cvars[i]=cs[i], i=1..ncs);
ceqs:= subs(cs, peqs);
sol:= solve(ceqs, bvars); svals:= map(rhs, sol);
if indets(svals) <> {} then error("Oops: %1", svals) fi;
if svals::set(nonnegint) and max(svals) <= 9 then
res:= res, [op(psol), cs, op(sol)];
fi
od
else
sol:= solve(peqs, bvars);
svals:= map(rhs, sol);
if indets(svals) <> {} then error("Oops: %1", svals) fi;
if svals::set(nonnegint) and max(svals) <= 9 then
res:= [op(psol), op(sol)];
fi
fi;
[res]
end proc:
G:= proc(d, n)
local eqs, i, rs, b0s;
eqs:= [d*b[0] - d - 10*c[0],
seq(d*b[i]+c[i-1] - b[n-i] - 10*c[i], i=1..n-2),
d*b[n-1] + c[n-2] - b[1] - 10*b[0]];
b0s:= [msolve(eqs[1] mod 10, 10)];
rs:= select(t -> (map(rhs, t))::set(nonnegint),
map(t -> t union solve(eval(eqs[1], t), {c[0]}), b0s));
for i from 1 to floor(n/2) do
rs:= map(s -> op(extend(d, s, {eqs[i+1], eqs[-i]})), rs);
od;
sort(map(s -> d*10^n + subs(s, add(10^i*b[i], i=0..n-1)), rs));
end proc:
A:= NULL;
for n from 2 to N-1 do
for d from 3 to 9 do
res:= G(d, n);
if res <> [] then
A:= A, op(res);
fi
od
od:
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