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%I #10 Apr 19 2019 11:00:33
%S 13731373,31373137,1190911909,9091190911,19090911909091,
%T 7316763215964848081373167632159648480813,
%U 111272689909091345969111272689909091345969,111279090913268945969111279090913268945969,112726894596919090913112726894596919090913
%N Subsequence of A083359 (Visual Factor Numbers) of the form (10^m+1)*p, where the decimal representation of prime p contains all the prime factors of 10^m+1.
%C Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).
%C The prime factors of 10^m + 1 are also prime factors of 10^km + 1, where k is odd, so a(n) and a(km) have those prime factors in common.
%C Expanding on the comment in A083359 regarding finding large terms, we can generate large doublet terms in the following way:
%C -- for any composite number M = 10^m + 1 try to compose a prime number p of length m from the prime factors of M. Generally this will require the factors to overlap to reduce the length to m.
%C -- the factors can wrap around to the beginning of p. For example, if M has a factor of 137 then p can be of the form 7...13.
%C -- the term is formed by concatenating p with itself to form a(n) = p||p. The resulting number will consist entirely of the concatenation of its prime factors with allowed overlap as in A083359.
%e With m = 4: 10^4 + 1 = 10001 = 73 * 137. We can form prime p = 1373 which concatenates with itself to give a(1) = 13731373 = 73 * 137 * 1373. We can also form the prime p = 3137 which gives a(2). The number 7313 also contains all the prime factors of 10001 but it is not prime.
%e With m = 33: 10^33 + 1 = 7*11*11*13*23*4093*8779*599144041*183411838171, there are 4932 m-digit numbers that contain all the factors, of which 227 of them are prime. Each of these primes generates a term in the sequence with 66 digits, the smallest of which is 112359914404134093877918341183817112359914404134093877918341183817. This is A324262(33).
%Y Cf. A083359, A020338, A324262, A324257.
%K nonn,base
%O 1,1
%A _Deron Stewart_, Mar 13 2019
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