

A068994


Powers of 2 with all even digits.


2




OFFSET

1,1


COMMENTS

Are there any more terms in this sequence?
Evidence that the sequence may be finite, from Rick L. Shepherd, Jun 23 2002:
1) The sequence of last two digits of 2^n, A000855 of period 20, makes clear that 2^n > 4 must have n == 3, 6, 10, 11, or 19 (mod 20) for 2^n to be a member of this sequence. Otherwise, either the tens digit (in 10 cases), as seen directly, or the hundreds digit, in the 5 cases receiving a carry from the previous power's tens digit >= 5, must be odd.
2) No additional term has been found for n up to 50000.
3) Furthermore, again for each n up to 50000, examining 2^n's digits leftward from the rightmost but only until an odd digit was found, it was only once necessary to search even to the 18th digit. This occurred for 2^12106 whose last digits are ...3833483966860466862424064. Note that 2^12106 has 3645 digits. (The clear runnerup, 2^34966, a 10526digit number, required searching only to the 15th digit. Exponents for which only the 14th digit was reached were only 590, 3490, 8426, 16223, 27771, 48966 and 49519  representing each congruence above.)
No additional terms up to 2^100000.  Harvey P. Dale, Dec 25 2012


LINKS

Table of n, a(n) for n=1..5.
Index to divisibility sequences


MATHEMATICA

(*returns true if none of digits of n are odd, false o.w.*) f[n_] := Module[{ a, l, r, i}, a = IntegerDigits[n]; l = Length[a]; r = True; For[i = 1, i <= l, i++, If[Mod[a[[i]], 2] == 1, r = False; Break[ ]]]; r] (*main routine*) Do[p = 2^i; If[f[p], Print[p]], {i, 1, 10^4}]
Select[2^Range[0, 100], Union[Take[DigitCount[#], {1, 1, 2}]]=={0}&] (* Harvey P. Dale, Dec 25 2012 *)
Select[2^Range[0, 100], AllTrue[IntegerDigits[#], EvenQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 18 2016 *)


PROG

(PARI) f(n)=n=vecsort(eval(Vec(Str(n)))%2, , 8); #v==1&&v[1]==0
m=Mod(1, 10^19); for(n=1, 1e5, m*=2; if(f(lift(m))&&f(2^n), print1(2^n", "))) \\ Charles R Greathouse IV, Apr 09 2012


CROSSREFS

Cf. A000855 (final two digits of 2^n), A096549.
Sequence in context: A065549 A067507 A320898 * A167182 A058345 A093843
Adjacent sequences: A068991 A068992 A068993 * A068995 A068996 A068997


KEYWORD

base,nonn


AUTHOR

Joseph L. Pe, Mar 14 2002


STATUS

approved



