A324656 a(n) is the number of successive primorials A002110(i) larger than n that need to be tried before sum n + A002110(i) is found to be composite.

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

Offset

1,1

Comments

a(n) = 0 if n + A002110(A235224(n)), i.e., n plus {the least primorial > n} is composite.

a(n) = 1 if n + A002110(A235224(n)) is prime, but n + A002110(1+A235224(n)) is composite.

a(n) = k if n + A002110(j+A235224(n)) is prime for j=0..k-1, but n + A002110(k+A235224(n)) is composite.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to primorial base

Index entries for sequences related to primorial numbers

Example

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 try to add (to the original n, which is 1) the next larger primorial, which is 6, and 7 is also prime, as are also 31, 211 and 2311. Only with A002110(6), 30030 + 1 is not a prime, thus a(1) = 5.
For n=3, the next larger primorial is 6, but 3+6 = 9 is composite, thus a(3) = 0.
For n=29, which is prime, we try adding it to four successively larger primorial numbers 30, 210, 2310, 30030, until we find 510510 which gives sum 510539 which is composite, thus a(29) = 4. In primorial base (A049345), 29 is written as 421 and the successive sums tested are: 1421, 10421, 100421, 1000421 and 10000421.
For n=121, which is not prime, but 210+121 = 331 is, while 2310+121 = 2431 is not, a(121) = 1.

Prog

(PARI)
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(k+b)), k++); (k); };

Crossrefs

Cf. A002110, A049345 (A324550), A235224, A324657.

Cf. also A005235, A324642.

Sequence in context: A368864 A368863 A090750 * A325817 A325967 A229656

Adjacent sequences: A324653 A324654 A324655 * A324657 A324658 A324659

Keyword

nonn

Author

Antti Karttunen, Mar 11 2019

Status

approved