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A358548
a(n) = A003627(n+1) - A003627(n).
3
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
1,1
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
LINKS
EXAMPLE
For n=3, a(3) = A003627(4) - A003627(3) = 17 - 11 = 6.
MATHEMATICA
A007528 := Select[6 Range[10000] - 1, PrimeQ]; m := Table[A007528[[n+1]] - A007528[[n]], {n, 1, 3040}]; PrependTo[m, 3] - Timothy L. Tiffin, Jan 5 2023
Differences[Select[Prime[Range[200]], Mod[#, 3]==2&]] (* Harvey P. Dale, Jan 31 2023 *)
PROG
(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
CROSSREFS
Sequence in context: A327576 A331058 A040006 * A155067 A094011 A351116
KEYWORD
nonn,easy
AUTHOR
Jordan Gunter, Nov 19 2022
STATUS
approved