cp's OEIS Frontend

A126359 A sequence of prime numbers that can be expressed using only digits 0 and 1 in minimum ascending bases.

%I A126359 #28 Jul 09 2025 04:26:17
%S A126359 3,5,7,37,43,223,1297,1303,1549,2801,4673,6571,10111,101111
%N A126359 A sequence of prime numbers that can be expressed using only digits 0 and 1 in minimum ascending bases.
%C A126359 3 = 10 base 3 = 3 + 0
%C A126359 5 = 10 base 5 = 5 + 0
%C A126359 7 = 11 base 6 = 6 + 1
%C A126359 37 = 101 base 6 = 36 + 0 + 1
%C A126359 43 = 111 base 6 = 36 + 6 + 1
%C A126359 223 = 1011 base 6 = 216 + 0 + 6 + 1
%C A126359 1297 = 10001 base 6 = 1296 + 0 + 0 + 0 + 1
%C A126359 1303 = 10011 base 6 = 1296 + 0 + 0 + 6 + 1
%C A126359 1549 = 11101 base 6 = 1296 + 216 + 36 + 0 + 1
%C A126359 2801 = 11111 base 7 = 2401 + 343 + 49 + 7 + 1
%C A126359 4673 = 11101 base 8 = 4096 + 512 + 64 + 0 + 1
%C A126359 6571 = 10011 base 9 = 6561 + 0 + 0 + 9 + 1
%C A126359 10111 = 10111 base 10 = 10000 + 0 + 100 + 10 + 1
%C A126359 101111 = 101111 base 10 = 100000 + 0 + 1000 + 100 + 10 + 1
%F A126359 Step 1: Starting at the first prime number (3), convert to the minimum base (3, as all primes may be expressed in binary).
%F A126359 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 A126359 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 A126359 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 A126359 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 A126359 A215511
%K A126359 nonn,base
%O A126359 3,1
%A A126359 _Jason Betts_, Aug 14 2012