A391796 a(n) is the least k such that the residues mod 3 of the primes prime(k), prime(k+1), ... comprise a string of n 2's followed by 1.

3, 9, 15, 54, 290, 987, 4530, 21481, 58554, 60967, 136457, 136456, 673393, 1254204, 1254203, 7709873, 21357254, 21357253, 25813465, 25813464, 39500858, 39500857, 947438661, 947438660, 947438659, 5703167679, 5703167678, 16976360924, 68745739764, 117327812949, 198156183037

Offset

1,1

Links

Table of n, a(n) for n=1..31.

Example

The primes mod 3 are A039701 = (2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2,...), in which (2,1) first appears in position 3, and (2,2,1) first occurs in position 9.

Mathematica

t = Table[Mod[Prime[nn], 3], {nn, 10^8}]; (* A039701 *)
u = Map[SequencePosition[t, Flatten[Join[{ConstantArray[2, #], 1}]], 1] &, Range[21]]
Table[First[Flatten[u[[n]]]], {n, 1, 21}]
(* Peter J. C. Moses, Dec 11 2025 *)

Prog

(Python)
from sympy import nextprime
from itertools import count, islice
def agen(NN=10000): # generator of terms for n <= NN
    n, pk, residues = 1, 1, "X"*NN
    for k in count(1):
        pk = nextprime(pk)
        resk = pk%3
        if resk == 1:
            while residues.endswith("2"*n) and n <= NN:
                yield k-n
                n += 1
        residues = residues[1:] + str(resk)
print(list(islice(agen(), 12))) # Michael S. Branicky, Jan 11 2026

Crossrefs

Cf. A390511, A391807, A392104.

Sequence in context: A329420 A120403 A247967 * A062804 A110960 A192165

Adjacent sequences: A391793 A391794 A391795 * A391797 A391798 A391799

Keyword

nonn

Author

Clark Kimberling, Jan 05 2026

Extensions

a(22)-a(27) from Michael S. Branicky, Jan 11 2026

a(28)-a(31) from Jinyuan Wang, Jan 16 2026

Status

approved