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

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

Offset: 1

Author

Graham Holmes, Aug 02 2020

Keywords

• nonn
• base
• fini
• full

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_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

No more terms beyond a(40). - Bert Dobbelaere, Sep 26 2020

Examples

			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).
		

Links

• David A. Corneth, PARI program

Crossrefs

Cf. A336768 (for bases >= 4).

Programs

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);

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 *)

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

Extensions

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