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 A083677 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. 7
 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, 337, 191, -1, 105, 88, 19, 28, 111, -1, 525, 13, 24, 102, 159, -1, 288, 142, 31, 743, 81, -1, 183, 202, 100, 96, 380, -1, 1227, 5, 113, 123, 20, 23, 0, 48, 148, 438, 52, 144, 128, 297, 206 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS C. Rivera, On Solution Of Puzzle 208 . EXAMPLE a(4) = 1 because 2030507 is composite and 2131517 is prime. MATHEMATICA fpkQ[k_, n_] := PrimeQ[ FromDigits[ Flatten[ IntegerDigits /@ Insert[ Table[ Prime[i], {i, k}], n, Table[{i}, {i, 2, k}]]]]]; a = 0; a = a = a = a = a = a = a = a = a = a = a = a = a = -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 *) CROSSREFS A082549 gives the n such that a(n) = 0. A083684 gives the n such that a(n)=-1. Cf. A082549, A032711, A083966, A083969, A092117, A090529. Sequence in context: A094424 A265241 A166888 * A075803 A127966 A326570 Adjacent sequences:  A083674 A083675 A083676 * A083678 A083679 A083680 KEYWORD sign,base AUTHOR Farideh Firoozbakht, Jun 15 2003 EXTENSIONS Edited and extended by Robert G. Wilson v, Dec 11 2004 STATUS approved

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)