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 A080439 a(1) = 11, a(n) = smallest prime obtained by inserting digits between every pair of digits of a(n-1). 4

%I

%S 11,101,10061,100000651,10000000000060571,

%T 100000000000000000000000600052761,

%U 10000000000000000000000000000000000000000000000060000000502271641

%N a(1) = 11, a(n) = 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 (amarnath_murthy(AT)yahoo.com), Feb 22 2003

%E Edited, corrected and extended by _Robert G. Wilson v_, Feb 22 2003

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