Jordan Gunter - cp's OEIS frontend

User: Jordan Gunter

Jordan Gunter has authored 2 sequences.

A358565 a(n) = A358548(n) / 6.

1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 3, 1, 1, 2, 1, 5, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 5, 1, 1, 4, 1, 2, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 6, 1, 1, 3, 1, 1, 3, 4, 1, 1, 1, 3, 1, 3, 3, 4, 3, 2, 4, 2, 2, 1, 2, 3, 1, 3, 1, 4, 3, 2, 1, 1, 3, 1, 1, 5, 1

Offset: 2

Jordan Gunter, Nov 22 2022

nonn

(1/6) * (first differences of consecutive primes of the form 3*k - 1).

Cf. A358548, A003627.

from itertools import islice
from sympy import isprime
def A358565_gen(): # generator of terms
    p, q = 5, 11
    while True:
        while not isprime(q):
            q += 3
        yield (q-p)//6
        p = q
        q += 3
A358565_list = list(islice(A358565_gen(),30)) # Chai Wah Wu, Jan 05 2023

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

Original entry on oeis.org

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

Jordan Gunter, Nov 19 2022

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

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

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

Cf. A003627, A359257.

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 *)

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

cp's OEIS Frontend

User: Jordan Gunter

A358565 a(n) = A358548(n) / 6.

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