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%I #7 Aug 19 2012 10:41:14
%S 10,10,11,101,111,1011,10001,10011,11101,11111,11101,10011,10111,
%T 101111
%N A sequence of prime numbers expressed as minimum bases using only digits 0 and 1.
%C 3 = 10 base 3 = 3 + 0
%C 5 = 10 base 5 = 5 + 0
%C 7 = 11 base 6 = 6 + 1
%C 37 = 101 base 6 = 36 + 0 + 1
%C 43 = 111 base 6 = 36 + 6 + 1
%C 223 = 1011 base 6 = 216 + 0 + 6 + 1
%C 1297 = 10001 base 6 = 1296 + 0 + 0 + 0 + 1
%C 1303 = 10011 base 6 = 1296 + 0 + 0 + 6 + 1
%C 1549 = 11101 base 6 = 1296 + 216 + 36 + 0 + 1
%C 2801 = 11111 base 7 = 2401 + 343 + 49 + 7 + 1
%C 4673 = 11101 base 8 = 4096 + 512 + 64 + 0 + 1
%C 6571 = 10011 base 9 = 6561 + 0 + 0 + 9 + 1
%C 10111 = 10111 base 10 = 10000 + 0 + 100 + 10 + 1
%C 101111 = 101111 base 10 = 100000 + 0 + 1000 + 100 + 10 + 1
%F Step 1: Starting at the first prime number (3), convert to the minimum base (3, as all primes may be expressed in binary).
%F Step 2: If the next prime number can be converted into the same base using only 0 and 1 without exceeding the value of the next prime number in the next base, this is the next item in the sequence.
%F Step 3: If the next prime number cannot be expressed in this base before exceeding the value of the next prime number in the next base, skip this prime number and move on to the next prime number and repeat Step 2.
%F Step 4: If the next prime number cannot be expressed in this base before exceeding the value of the next prime number in the next base, but can be expressed in the next base, this is the next item in the sequence.
%e The first term is 3 in base 3. The next prime in that base is 13, which is greater than the value of the prime in the next base, which is 5 in base 4, so the second term is 5 in base 4.
%Y A126359
%K nonn,base
%O 3,1
%A _Jason Betts_, Aug 14 2012.
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