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%I #9 Dec 23 2021 02:25:19
%S 17,107,10007,100000007,10000000000003037,
%T 100000000000000000000000003000307,
%U 10000000000000000000000000000000000000000000000000003000000030057
%N a(1) = 17, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).
%C Conjecture: Only one digit needs to be inserted between each pair of digits of a(n-1) to get a(n); i.e. a(n) contains exactly 2n-1 digits for n > 1.
%C The conjecture above is false: a(5) = 10000000000003037 has 17 digits instead of 2*5 - 1 = 9. A refined conjecture is: a(n) contains exactly 2^(n-1) + 1 digits for all n > 0. This follows trivially by induction from the initial conjecture above of only one digit needed between each pair, and the fact that we start with 17, a 2-digit number, and holds true at least till a(12).
%H Julio Cesar Hernandez-Castro, <a href="/A080441/b080441.txt">Table of n, a(n) for n = 1..12</a>
%t a[n_] := Block[{d = IntegerDigits[n]}, k = Length[d]; While[k > 1, d = Insert[d, 0, k]; k-- ]; d = FromDigits[d]; e = d; k = 0; While[ !PrimeQ[e], k++; e = d + 10FromDigits[ IntegerDigits[k], 100]]; e]; NestList[a, 17, 6]
%Y Cf. A080439, A080440, A080442, A080883 - A080914.
%K base,nonn
%O 1,1
%A _Amarnath Murthy_, Feb 22 2003
%E Edited and extended by _Robert G. Wilson v_, Feb 22 2003
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