A080440 - OEIS

%I #13 Dec 23 2021 02:24:22

%S 13,103,10093,100000963,10000000000092653,

%T 100000000000000000000000902060523,

%U 10000000000000000000000000000000000000000000000090002000600051233

%N a(1) = 13, 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(4) = 100000963 has 9 digits instead of 2*4 - 1 = 7. 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 above conjecture of only one digit needed between each pair, and the fact that we start with 13, a 2-digit number, and holds true at least till a(12). - _Julio Cesar Hernandez-Castro_, Jul 06 2011

%H Julio Cesar Hernandez-Castro, <a href="/A080440/b080440.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, 13, 6]

%Y Cf. A080439, A080441, A080442, A080883 - A080914.

%K base,nonn

%O 1,1

%A _Amarnath Murthy_, Feb 22 2003

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