

A336733


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


3



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

1,2


COMMENTS

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 (base 10) = 240 (base 6)) will see that zero preserved when the string of digits is reversed (e.g. into "042"), so it can never match any baseconverted 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 baseconverted strings to be of the same length; considering that any solution for a sufficientlyhigh 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 (base 10) = 211 (base 3) = 112 (base 4), and 22 (base 10) = 42 (base 5) = 24 (base 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


LINKS

Table of n, a(n) for n=1..40.
David A. Corneth, PARI program


EXAMPLE

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).


MATHEMATICA

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 *)


PROG

(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


CROSSREFS

Cf. A336768 (for bases >= 4).
Sequence in context: A320318 A005577 A263362 * A072966 A245761 A059759
Adjacent sequences: A336730 A336731 A336732 * A336734 A336735 A336736


KEYWORD

nonn,base,hard,more,fini,changed


AUTHOR

Graham Holmes, Aug 02 2020


EXTENSIONS

a(40) from David A. Corneth, Aug 07 2020


STATUS

approved



