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A267939 Number x = concat(MSD(x),b), where MSD = A000030 stands for Most Significant Digit, such that MSD(x)*b is equal to the reverse of x. 3

%I #31 Feb 04 2016 09:57:45

%S 351,621,886,920781,3524751,338752611,35247524751,920780120781,

%T 920879219781,3387524752611,3526124738751,338738752612611,

%U 352475247524751,33875247524752611,35247387526124751,35261247524738751,920780120780120781,920780219879120781,920879120780219781,920879219879219781

%N Number x = concat(MSD(x),b), where MSD = A000030 stands for Most Significant Digit, such that MSD(x)*b is equal to the reverse of x.

%C If we consider numbers x = concat(a,b), where a has two digits, such that a*b is equal to the reverse of x, the first terms are 425322, 44235301, 119910901, ...

%C Terms of the form 3(5247)*51, i.e. 351, 3524751, 35247524751, ..., form an infinite subsequence. - _Robert Israel_, Jan 28 2016

%C Other infinite sequences of terms include 92078(012078)*1 and 33875(2475)*2611. - _Robert Israel_, Jan 31 2016

%H Robert Israel, <a href="/A267939/b267939.txt">Table of n, a(n) for n = 1..415</a>

%e 3*51 = 153;

%e 6*21 = 126;

%e 3*524751 = 1574253.

%p T:=proc(w) local x, y, z; x:=w; y:=0;

%p for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:

%p P:=proc(q) local a,b,n; for n from 1 to q do a:=n mod 10; b:=trunc(n/10^ilog10(n));

%p 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

%p if T(n)=b*(n mod 10^ilog10(n)) then print(n); fi; fi; od; end: P(10^10);

%p # alternative:

%p N:= 20: # to get all terms with at most N digits.

%p extend:= proc(d,psol,eqs)

%p local peqs, cvars, bvars, ncs, res,T, cs, ceqs, sol, svals;

%p peqs:= subs(psol, eqs);

%p cvars,bvars:= selectremove(t -> op(0,t) = 'c',indets(peqs));

%p ncs:= nops(cvars);

%p res:= NULL;

%p if ncs >= 1 then

%p T:= combinat:-cartprod([[$0..d-1]$ncs]);

%p while not T[finished] do

%p cs:= T[nextvalue]();

%p cs:= seq(cvars[i]=cs[i],i=1..ncs);

%p ceqs:= subs(cs,peqs);

%p sol:= solve(ceqs,bvars); svals:= map(rhs,sol);

%p if indets(svals) <> {} then error("Oops: %1",svals) fi;

%p if svals::set(nonnegint) and max(svals) <= 9 then

%p res:= res, [op(psol), cs, op(sol)];

%p fi

%p od

%p else

%p sol:= solve(peqs,bvars);

%p svals:= map(rhs,sol);

%p if indets(svals) <> {} then error("Oops: %1",svals) fi;

%p if svals::set(nonnegint) and max(svals) <= 9 then

%p res:= [op(psol), op(sol)];

%p fi

%p fi;

%p [res]

%p end proc:

%p G:= proc(d,n)

%p local eqs, i, rs, b0s;

%p eqs:= [d*b[0] - d - 10*c[0],

%p seq(d*b[i]+c[i-1] - b[n-i] - 10*c[i], i=1..n-2),

%p d*b[n-1] + c[n-2] - b[1] - 10*b[0]];

%p b0s:= [msolve(eqs[1] mod 10,10)];

%p rs:= select(t -> (map(rhs,t))::set(nonnegint),

%p map(t -> t union solve(eval(eqs[1],t),{c[0]}),b0s));

%p for i from 1 to floor(n/2) do

%p rs:= map(s -> op(extend(d,s,{eqs[i+1],eqs[-i]})), rs);

%p od;

%p sort(map(s -> d*10^n + subs(s, add(10^i*b[i],i=0..n-1)), rs));

%p end proc:

%p A:= NULL;

%p for n from 2 to N-1 do

%p for d from 3 to 9 do

%p res:= G(d,n);

%p if res <> [] then

%p A:= A, op(res);

%p fi

%p od

%p od:

%p A; # _Robert Israel_, Feb 01 2016

%t Select[Range@ 4000000, First[#] FromDigits@ Rest@ # == FromDigits@ Reverse@ # &@ IntegerDigits@ # &] (* _Michael De Vlieger_, Jan 29 2016 *)

%Y Cf. A000030, A004086.

%K base,nonn

%O 1,1

%A _Paolo P. Lava_, Jan 22 2016

%E a(7) to a(20) from _Robert Israel_, Feb 01 2016

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