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%I #22 Dec 23 2021 00:50:54
%S 11,101,10061,100000651,10000000000060571,
%T 100000000000000000000000600052761,
%U 10000000000000000000000000000000000000000000000060000000502271641
%N a(1) = 11, 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)=10000000000060571 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 11, a 2-digit number, and holds true at least till a(12). - _Julio Cesar Hernandez-Castro_, Jul 05 2011
%H Julio Cesar Hernandez-Castro, <a href="/A080439/b080439.txt">Table of n, a(n) for n = 1..12</a>
%e a(2) = 101 and a(3) is obtained by inserting a '0' and a '6' in the two inner spaces of 101: (1,-,0,-,1).
%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, 11, 6]
%Y Cf. A080440, A080441, A080442, A080883 - A080914.
%K nonn,base
%O 1,1
%A _Amarnath Murthy_, Feb 22 2003
%E Edited, corrected and extended by _Robert G. Wilson v_, Feb 22 2003
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