1,1

Are there any more terms in this sequence?

Evidence that the sequence may be finite, from Rick L. Shepherd (rshepherd2(AT)hotmail.com), 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.)

(*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}]

Cf. A000855 (final two digits of 2^n).

Sequence in context: A069027 A065549 A067507 this_sequence A167182 A058345 A093843

Adjacent sequences: A068991 A068992 A068993 this_sequence A068995 A068996 A068997

base,nonn

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Mar 14 2002