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A068994 Powers of 2 with all even digits. 2
2, 4, 8, 64, 2048 (list; graph; refs; listen; history; text; internal format)
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 runner-up, 2^34966, a 10526-digit 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

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Last modified November 22 03:20 EST 2019. Contains 329383 sequences. (Running on oeis4.)