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%I #2 Mar 30 2012 17:31:00
%S 1,3,9,13,63,107,27,67,39,23,49,29,99,439,207,41,357,229,77,139,69,
%T 839,133,239,121,317,187,53,33,1291,177,557,171,1753,323,19,519,953,
%U 231,523,321,251,327,31,299,2203,747,101,81,1741,291,6779,261,1549,1463,97,297
%N Alternate composite and prime numbers not included earlier such that every partial concatenation is a prime: a(2n) is prime and a(2n-1) is not prime.
%C Conjecture: 2 and 5 are the only two nonmembers.
%e 1, 13, 139, 13913, 1391363, 1391363107,..., etc. are not composite.
%t p = Prime[ Range[ 1500]]; np = Drop[ Complement[ Range[ 1500], p], 1]; a[1] = 1; a[n_] := a[n] = Block[{k = 1, q = Flatten[ IntegerDigits[ # ] & /@ Table[ a[i], {i, n - 1}]]}, If[ EvenQ[n], While[ !PrimeQ[ FromDigits[ Join[q, IntegerDigits[ p[[k]] ]]]], k++ ]; q = p[[k]]; p = Delete[p, k]; q, While[ !PrimeQ[ FromDigits[ Join[q, IntegerDigits[ np[[k]] ]]]], k++ ]; q = np[[k]]; np = Delete[np, k]; q]]; Table[ a[n], {n, 60}]
%Y Cf. A088614, A094045.
%K nonn,base
%O 1,2
%A _Robert G. Wilson v_, Apr 23 2004
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