cp's OEIS Frontend

A068994 Powers of 2 with all even digits.

%I A068994 #57 Apr 07 2025 00:03:21
%S A068994 2,4,8,64,2048
%N A068994 Powers of 2 with all even digits.
%C A068994 The corresponding exponents are: 1, 2, 3, 6, 11.
%C A068994 Are there any more terms in this sequence?
%C A068994 Evidence that the sequence may be finite, from _Rick L. Shepherd_, Jun 23 2002:
%C A068994 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.
%C A068994 2) No additional term has been found for n up to 50000.
%C A068994 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.)
%C A068994 No additional terms up to 2^100000. - _Harvey P. Dale_, Dec 25 2012
%C A068994 No additional terms up to 2^(10^10). - _Michael S. Branicky_, Apr 16 2023
%C A068994 No additional terms up to 2^(10^11). See C program in links. It was only necessary to check the rightmost 40 digits of each power. But for going towards 10^12, 40 digits won't suffice. - _Lukas Huwald_, Mar 20 2025
%C A068994 No additional terms up to 2^(10^13). See second C program in links. The champions in this range are 2^133477987019 with 46 even last digits and 2^9780164740006 with 45 even last digits. - _Fredrik Johansson_, Mar 21 2025
%C A068994 No additional terms up to 2^(10^17). See Go program in links which uses a strong sieve to accelerate the computation substantially. - _Ted E Dunning_, Mar 31 2025
%H A068994 Bogdan C. Dumitru, <a href="https://arxiv.org/abs/2503.23177">Finiteness of Powers of Two with All Even Digits</a>, arXiv:2503.23177 [math.NT], 2025.
%H A068994 Ted Dunning, <a href="https://github.com/tdunning/EvenDigits">Go program with sieve to accelerate scanning by ~1000x</a>.
%H A068994 Lukas Huwald, <a href="/A068994/a068994.c.txt">C program to check powers of 2 for even digits</a>.
%H A068994 Fredrik Johansson, <a href="https://gist.github.com/fredrik-johansson/8924e10e5d74e391094893932914c852">Second C program</a>.
%H A068994 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%t A068994 (*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}]
%t A068994 Select[2^Range[0,100],Union[Take[DigitCount[#],{1,-1,2}]]=={0}&] (* _Harvey P. Dale_, Dec 25 2012 *)
%t A068994 Select[2^Range[0,100],AllTrue[IntegerDigits[#],EvenQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Sep 18 2016 *)
%o A068994 (PARI) f(n)=n=vecsort(eval(Vec(Str(n)))%2,,8);#v==1&&v[1]==0
%o A068994 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
%Y A068994 Cf. A000855 (final two digits of 2^n), A096549.
%K A068994 base,nonn
%O A068994 1,1
%A A068994 _Joseph L. Pe_, Mar 14 2002