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%I #12 Mar 28 2015 18:09:08
%S 0,2,-1,1,4,10,38,20,0,-1,163,46,8,53,0,-1,74,5,8,5,180,4,280,191,0,
%T 337,191,-1,105,88,19,28,111,-1,525,13,24,102,159,-1,288,142,31,743,
%U 81,-1,183,202,100,96,380,-1,1227,5,113,123,20,23,0,48,148,438,52,144,128,297,206
%N Define f(n, k) to be the concatenation of the first n primes, with n-1 k's inserted between the primes. Then a(n) is the smallest k >= 0 such that f(n, k) is prime, or -1 if no such prime exists.
%C a(3) = -1 because f(3, k) is always a multiple of 5. For any n such that n = 1 (mod 3) and A007504(n) = 0 (mod 3), a(n) = -1 because f(n, k) is always a multiple of 3. It is my conjecture that for all other n, -1 < a(n) < n*p(n). I've checked for all n < 270.
%H C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_208.htm">On Solution Of Puzzle 208 </a>.
%e a(4) = 1 because 2030507 is composite and 2131517 is prime.
%t fpkQ[k_, n_] := PrimeQ[ FromDigits[ Flatten[ IntegerDigits /@ Insert[ Table[ Prime[i], {i, k}], n, Table[{i}, {i, 2, k}]]]]]; a[1] = 0; a[3] = a[10] = a[16] = a[28] = a[34] = a[40] = a[46] = a[52] = a[70] = a[76] = a[82] = a[88] = a[97] = -1; a[n_] := Block[{k = 0}, While[ fpkQ[n, k] != True, k++ ]; k]; Table[ a[n], {n, 70}] (* _Robert G. Wilson v_, Dec 11 2004 *)
%Y A082549 gives the n such that a(n) = 0. A083684 gives the n such that a(n)=-1.
%Y Cf. A082549, A032711, A083966, A083969, A092117, A090529.
%K sign,base
%O 1,2
%A _Farideh Firoozbakht_, Jun 15 2003
%E Edited and extended by _Robert G. Wilson v_, Dec 11 2004
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