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%I #26 Aug 19 2012 10:35:18
%S 3,5,7,37,43,223,1297,1303,1549,2801,4673,6571,10111,101111
%N A sequence of prime numbers that can be expressed using only digits 0 and 1 in minimum ascending bases.
%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 2801 is a prime in this sequence in base 7. The next prime in base 7 is 17207 but this exceeds the value of a prime in base 8, such that A(n) (base y) < A(n+1) (base y+1) < A(n+1) (base y), so the next number in this sequence must go to the next base 8, which is 4673, because the number after 2081 in base 7 is 17207, and 4673 < 17207.
%Y A215511
%K nonn,base
%O 3,1
%A _Jason Betts_, Aug 14 2012.
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