A068588 Treated as strings, n and its reversal are substrings of 2^n.

6, 10, 44, 49, 60, 67, 190, 191, 226, 252, 321, 373, 430, 521, 551, 609, 613, 660, 666, 680, 697, 703, 727, 730, 742, 750, 777, 805, 810, 838, 842, 847, 851, 861, 879, 889, 905, 913, 919, 920, 944, 949, 950, 959, 968, 973, 982

Offset: 1

Author

Joseph L. Pe, Mar 27 2002

Keywords

• nonn
• base
• changed

Comments

For an arithmetical function f whose range is a set of integers, call n a "fixated point" of f if, treated as strings, n and its reversal are both substrings of f(n). a(n) above lists the fixated points of f(n) = 2^n. In general, the faster f(n) grows with respect to n, the more digits f(n) will have as compared with n, hence the likelier it is to contain n and n' as substrings. Thus it is more interesting to consider f(n) which grows slowly with respect to n. One such function is given by f(n) = Prime(n). The only fixated points of f not exceeding 107 are 7 and 6460. Are there any more such points?

Examples

			2^49 = 562949953421312 in which both 49 and its reversal 94 appear as substrings, so 49 belongs to the sequence.
		

Links

• Joseph L. Pe, Fixated Points of Arithmetical Functions.

Programs

Mathematica

Do[m = 2^n; If[StringPosition[ToString[m], ToString[n]] != {} && StringPosition[ToString[m], ToString[FromDigits[Reverse[IntegerDigits[n]]]]] != {}, Print[n]], {n, 1, 1000}]
sbsQ[n_]:=Module[{c=IntegerDigits[2^n]},SequenceCount[c,IntegerDigits[n]]>0 && SequenceCount[c,IntegerDigits[IntegerReverse[n]]]>0]; Select[Range[1000],sbsQ] (* Harvey P. Dale, May 08 2022 *)