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.

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

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

Table of n, a(n) for n=1..60.

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