|
%I #8 May 08 2022 18:39:43
%S 6,10,44,49,60,67,190,191,226,252,321,373,430,521,551,609,613,660,666,
%T 680,697,703,727,730,742,750,777,805,810,838,842,847,851,861,879,889,
%U 905,913,919,920,944,949,950,959,968,973,982
%N Treated as strings, n and its reversal are substrings of 2^n.
%C 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?
%H Pe, J., <a href="http://numeratus.net/enlightened/fixated.html">Fixated Points of Arithmetical Functions</a> [Link updated by Jason G. Wurtzel, Sep 07 2010]
%e 2^49 = 562949953421312 in which both 49 and its reversal 94 appear as substrings, so 49 belongs to the sequence.
%t Do[m = 2^n; If[StringPosition[ToString[m], ToString[n]] != {} && StringPosition[ToString[m], ToString[FromDigits[Reverse[IntegerDigits[n]]]]] != {}, Print[n]], {n, 1, 1000}]
%t 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 *)
%K nonn,base
%O 1,1
%A _Joseph L. Pe_, Mar 27 2002
|
