A324550 Primes written in primorial base (A049345).

10, 11, 21, 101, 121, 201, 221, 301, 321, 421, 1001, 1101, 1121, 1201, 1221, 1321, 1421, 2001, 2101, 2121, 2201, 2301, 2321, 2421, 3101, 3121, 3201, 3221, 3301, 3321, 4101, 4121, 4221, 4301, 4421, 5001, 5101, 5201, 5221, 5321, 5421, 6001, 6121, 6201, 6221, 6301, 10001, 10201, 10221, 10301, 10321, 10421, 11001, 11121, 11221

Offset: 1

Author

Antti Karttunen, Mar 11 2019

Keywords

• nonn
• base

Comments

When the primorial base representation is expressed with decimal digits as here, the sequence stays unambiguous only up to the 317th prime, 2099, written as 96421, because after that primorial base digits larger than 9 would be needed.

By writing down terms from a(6) to a(46) (primes 13 .. 199):

201, 221, 301, 321, 421, 1001, 1101, 1121, 1201, 1221, 1321, 1421, 2001, 2101, 2121, 2201, 2301, 2321, 2421, 3101, 3121, 3201, 3221, 3301, 3321, 4101, 4121, 4221, 4301, 4421, 5001, 5101, 5201, 5221, 5321, 5421, 6001, 6121, 6201, 6221, 6301,

and then from a(48) to a(80) (primes 223 .. 409):

10201, 10221, 10301, 10321, 10421, 11001, 11121, 11221, 11321, 11421, 12001, 12101, 12121, 12201, 12321, 13101, 13121, 13201, 13221, 14001, 14101, 14221, 14301, 14321, 14421, 15101, 15201, 15301, 15321, 15421, 16101, 16121, 16301,

it is clearly seen that if n is a prime, then p+n is also likely to be prime, where p is the next higher primorial (A002110) > n. See also A324656.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..317
• Index entries for sequences related to primorial base.

Crossrefs

Cf. A000040, A018239, A049345, A324642, A324656, A324657.

Cf. also A004676, A214617, A286941.

Programs

Mathematica

a[n_] := Module[{k = Prime[n], p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; FromDigits[Reverse[s]]]; Array[a, 100] (* Amiram Eldar, Mar 06 2024 *)

PARI

A324550(n) = A049345(prime(n)); \\ For A049345, see under that entry.

Formula

a(n) = A049345(A000040(n)).