A324656 -id:A324656 - cp's OEIS frontend

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

Antti Karttunen, Mar 11 2019

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.

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

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

Cf. also A004676, A214617, A286941.

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 *)

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

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

A324642 Number of iterations of map x -> x + A002110(A235224(x)) required to reach a composite when starting from x = n. Here A002110(A235224(x)) gives the least primorial number > x.

Original entry on oeis.org

2, 1, 1, 0, 4, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 3, 0, 2, 0, 0

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

			For n=1, it is not a composite number, so we add a next larger primorial (A002110) to it, which is 2, and we see that 3 is also noncomposite, thus we add to that the next larger primorial, which is 6, but now 3+6 = 9 is composite, which we reached in two iteration steps, thus a(1) = 2.
For n = 97, the iteration goes as: 97 -> 307 -> 2617 -> 32647 -> 543157 -> 10242847 -> 233335717 -> 6703028947 -> 207263519077, and only the last term shown is composite, thus a(97) = 8. Written in primorial base (A049345), the terms in that trajectory look as: 3101, 13101, 113101, 1113101, 11113101, 111113101, 1111113101, 11111113101 and 111111113101.

Cf. A002110, A049345 (A324550), A235224.

Cf. also A063377, A324656.

A002110(n) = prod(i=1,n,prime(i));
A235224(n, p=2) = if(nA235224(n\p, nextprime(p+1)));
A324642(n) = { my(k=0); while((1==n)||isprime(n), n += A002110(A235224(n)); k++); (k); };

If n is composite, a(n) = 0, and for noncomposite n, a(n) = 1 + a(n+A002110(A235224(n))).

A324657 a(n) is the number of successive primorials A002110(i) larger than prime(n) that need to be tried before sum prime(n)+A002110(i) is found to be composite.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 0, 3, 4, 2, 0, 5, 0, 2, 1, 1, 3, 3, 2, 5, 0, 3, 0, 1, 2, 1, 4, 0, 0, 2, 0, 2, 1, 2, 0, 3, 2, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 2, 1, 2, 3, 0, 0, 0, 1, 0, 0, 1, 4, 0, 1, 2, 0, 0, 0, 4, 1, 1, 1, 0, 1, 2, 1, 1, 1, 3, 2, 1, 2, 1, 1, 0, 2, 1, 0, 1, 0, 0, 2, 4, 2, 1, 0, 0, 3, 0, 4, 1, 3, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

See comments and examples in A324656.

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to primorial numbers

Cf. A000040, A002110, A324656.

A002110(n) = prod(i=1,n,prime(i));
A235224(n, p=2) = if(!n, n, 1+A235224(n\p, nextprime(p+1)));
A324656(n) = { my(k=0,b=A235224(n)); while(isprime(n+A002110(b+k)), k++); (k); };
A324657(n) = A324656(prime(n));

a(n) = A324656(A000040(n)).

cp's OEIS Frontend

A324550 Primes written in primorial base (A049345).

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A324642 Number of iterations of map x -> x + A002110(A235224(x)) required to reach a composite when starting from x = n. Here A002110(A235224(x)) gives the least primorial number > x.

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A324657 a(n) is the number of successive primorials A002110(i) larger than prime(n) that need to be tried before sum prime(n)+A002110(i) is found to be composite.

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