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A336733 Positive integers which can be written in two bases smaller than 10 as mutually-reversed strings of digit(s).

%I A336733 #85 Jun 28 2025 19:46:59
%S A336733 1,2,3,4,5,6,7,9,11,17,22,31,51,87,91,102,121,212,220,248,2601,5258,
%T A336733 7491,8283,9831,10516,13541,15774,16566,71500,644765,731445,811518,
%U A336733 3552340,314767045,1427310725,1848187230,1916060910,47124212513,455075911977
%N A336733 Positive integers which can be written in two bases smaller than 10 as mutually-reversed strings of digit(s).
%C A336733 Base conversion yields a string of digits which by convention has any leading zeros suppressed. However, a conversion which yields a low zero (e.g., 96_10 = 240_6) will see that zero preserved when the string of digits is reversed (e.g., into "042"), so it can never match any base-converted strings before reversal. It's therefore not possible to have a solution involving a base which exhibits a low zero for any input x. A consequence of this is that any solution will require both base-converted strings to be of the same length; considering that any solution for a sufficiently-high x will involve only bases 8 and 9 (these having the slowest rate of change with respect to x), we can deduce that the upper limit for valid solutions occurs at the point beyond which length(x base 8) - length(x base 9) is permanently greater than unity, and this can be shown to occur at 8^18, or approximately 1.80*10^16.
%C A336733 40 terms are known up to 4.7*10^13.
%C A336733 It's worthy of note that 22 has two distinct nontrivial solutions as 22_10 = 211_3 = 112_4, and 22_10 = 42_5 = 24_9.
%C A336733 As 1 through 6 have one digit in at least two distinct bases each less than 10 they are trivially included in the sequence. - _David A. Corneth_, Aug 03 2020
%C A336733 No more terms beyond a(40). - _Bert Dobbelaere_, Sep 26 2020
%H A336733 David A. Corneth, <a href="/A336733/a336733.gp.txt">PARI program</a>
%e A336733 7 is a term since 7 = 21 (base 3) = 12 (base 5).
%e A336733 9 is a term since 9 = 21 (base 4) = 12 (base 7).
%e A336733 ...
%e A336733 1916060910 is a term since it is = 65324151261 (base 7) = 16215142356 (base 8).
%t A336733 seqQ[n_] := Module[{dig = IntegerDigits[n, Range[2, 9]]}, dig = Select[dig, ! PalindromeQ[#] &]; n < 7 || Intersection[dig, Reverse /@ dig] != {}]; Select[Range[10^6], seqQ] (* _Amiram Eldar_, Aug 04 2020 *)
%o A336733 (JavaScript) n=[]; rev=[]; incl=[]; for (i=1; i<=1000; i++) { for (j=2; j<=9; j++) { n[j]=i.toString(j); rev[j]=n[j].split("").reverse().join(""); } for (j=2; j<=8; j++) for (k=j+1; k<=9; k++) if (n[j]==rev[k]) if (incl.indexOf(i)==-1) incl.push(i); } document.write(incl);
%o A336733 (PARI) isok(m) = {for (b=2, 8, my(db = digits(m, b)); for(c=b+1, 9, my(dc = digits(m, c)); if (Vecrev(dc) == db, return (1));););} \\ _Michel Marcus_, Aug 03 2020
%o A336733 (PARI) is(n) = {my(v = vecsort(vector(8, i, d = digits(n, i+1); if(d[1] < d[#d], Vecrev(d), d)))); for(i = 1, 7, if(v[i] == v[i+1], return(1))); 0} \\ _David A. Corneth_, Aug 03 2020
%Y A336733 Cf. A336768 (for bases >= 4).
%K A336733 nonn,base,fini,full
%O A336733 1,2
%A A336733 _Graham Holmes_, Aug 02 2020
%E A336733 a(40) from _David A. Corneth_, Aug 07 2020