A358548 a(n) = A003627(n+1) - A003627(n).

3, 6, 6, 6, 6, 12, 6, 6, 6, 12, 12, 6, 12, 6, 6, 18, 6, 12, 18, 6, 6, 12, 6, 30, 6, 6, 12, 6, 6, 6, 12, 12, 18, 6, 30, 6, 6, 24, 6, 12, 18, 12, 12, 6, 12, 6, 12, 12, 12, 6, 12, 36, 6, 6, 18, 6, 6, 18, 24, 6, 6, 6, 18, 6, 18, 18, 24, 18, 12, 24, 12, 12, 6, 12, 18, 6, 18, 6, 24

Offset: 1

Author

Jordan Gunter, Nov 19 2022

Keywords

• nonn
• easy

Comments

Sequence of differences between consecutive primes of the form 3n-1, which is the sequence A003627.

For n > 1, this is the sequence of first differences of A007528. The longest run of 6's will have length 4 (corresponding to 5, 11, 17, 23, 29 in A007528). Since no other prime ends in 5, thereafter, a run of 6's cannot exceed length 3 (e.g., 41, 47, 53, 59 in A007528). Similarly, a run of 12's cannot exceed length 3 (e.g., 467, 479, 491, 503 in A007528), a run of 18's cannot exceed length 3 (e.g., 2843, 2861, 2879, 2897 in A007528), and a run of 24's cannot exceed length 3 (e.g., 154619, 154643, 154667, 154691 in A007528). This run limit of length 3 also extends to other multiples of 6 that are not divisible by 5. - Timothy L. Tiffin, Dec 22 2022

For multiples of 6 that are divisible by 5, the length of the longest run does not appear to be bounded. For example, if k, k+30, k+60, k+90, ..., k+30(k-1) = 31k-30 are k consecutive primes in A002476, then this will produce a run of k-1 30's in this sequence. - Timothy L. Tiffin, Jan 06 2023

Examples

			For n=3, a(3) = A003627(4) - A003627(3) = 17 - 11 = 6.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003627, A359257.

Programs

Mathematica

A007528 := Select[6 Range[200] - 1, PrimeQ]; m := Table[A007528[[n+1]] - A007528[[n]], {n, 1, Length[A007528]-1}]; PrependTo[m, 3] (* Timothy L. Tiffin, Jan 05 2023 *)
Differences[Select[Prime[Range[200]],Mod[#,3]==2&]] (* Harvey P. Dale, Jan 31 2023 *)

Python

from itertools import islice
from sympy import isprime
def A358548_gen(): # generator of terms
    p, q = 2, 5
    while True:
        while not isprime(q):
            q += 3
        yield q-p
        p = q
        q += 3
A358548_list = list(islice(A358548_gen(),30)) # Chai Wah Wu, Jan 05 2023