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A336733
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Positive integers which can be written in two bases smaller than 10 as mutually-reversed strings of digit(s).
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3
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1, 2, 3, 4, 5, 6, 7, 9, 11, 17, 22, 31, 51, 87, 91, 102, 121, 212, 220, 248, 2601, 5258, 7491, 8283, 9831, 10516, 13541, 15774, 16566, 71500, 644765, 731445, 811518, 3552340, 314767045, 1427310725, 1848187230, 1916060910, 47124212513, 455075911977
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OFFSET
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1,2
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COMMENTS
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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.
40 terms are known up to 4.7*10^13.
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.
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
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LINKS
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EXAMPLE
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7 is a term since 7 = 21 (base 3) = 12 (base 5).
9 is a term since 9 = 21 (base 4) = 12 (base 7).
...
1916060910 is a term since it is = 65324151261 (base 7) = 16215142356 (base 8).
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MATHEMATICA
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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 *)
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PROG
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(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);
(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
(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
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CROSSREFS
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KEYWORD
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nonn,base,hard,fini,full
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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