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 A299154 a(n) is the distance from prime(n) to the next prime that has the same digital root, or a(n) = 0 if there is no greater prime with the same digital root. 0
 9, 0, 18, 36, 18, 18, 36, 18, 18, 18, 36, 36, 18, 18, 36, 18, 54, 18, 36, 18, 36, 18, 18, 18, 54, 36, 36, 72, 18, 18, 36, 18, 36, 18, 18, 72, 36, 18, 72, 18, 18, 18, 36, 18, 36, 72, 18, 18, 36, 54, 18, 18, 36, 18, 36, 18, 90, 36, 36, 36 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For n >= 3, a(n) is a multiple of 18. I conjecture that there are infinitely many prime numbers with each of the digital roots from the set {1,2,4,5,7,8}. This would imply that the only prime number with a(n) = 0 is 3. The conjecture is true; for any d in {1,2,4,5,7,8}, gcd(d, 9) = 1, hence, according to Dirichlet's theorem on arithmetic progressions, there are infinitely many primes of the form d + 9*k, and these primes all have digital root d. - Rémy Sigrist, Mar 25 2018 LINKS EXAMPLE For n=1, prime(n) = 2; 2 mod 9 = 2. The next prime that has the same digital root is 11, because 11 mod 9 = 2. So 11 - 2 = 9 is the first term. MATHEMATICA Table[With[{p = Prime@ n}, If[IntegerQ@ #, # - p, 0] &@ SelectFirst[Prime@ Range[n + 1, n + 120], SameQ @@ Mod[{#, p}, 9] &]], {n, 60}] (* Michael De Vlieger, Feb 10 2018 *) PROG (PARI) { print1(9", "0", "); forprime(n=5, 1000,          p=n%9; i=nextprime(n+1);          while((i%9)<>p, i=nextprime(i+1));          print1(i-n", ")         ) } CROSSREFS Cf. A038194. Sequence in context: A062047 A117465 A136679 * A070929 A007394 A067153 Adjacent sequences:  A299151 A299152 A299153 * A299155 A299156 A299157 KEYWORD nonn,base AUTHOR Dimitris Valianatos, Feb 03 2018 STATUS approved

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Last modified October 15 03:04 EDT 2019. Contains 328025 sequences. (Running on oeis4.)