A391807 a(n) is the least k such that the residues mod 3 of the primes prime(k), prime(k+1),... include a string of n 1's followed by 2.

4, 11, 36, 273, 272, 271, 2209, 11199, 13718, 13717, 34369, 172147, 172146, 3094796, 3094795, 4308948, 12762142, 23902561, 72084958, 72084957, 72084956, 1052779162, 1052779161, 1857276774, 1857276773, 19398320447, 57446769095, 57446769094, 57446769093, 57446769092, 57446769091

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 (1,2) first appears in position 4, and (1,1,2) first occurs in position 11, so a(1) = 4 and a(2) = 11.

Mathematica

t = Table[Mod[Prime[nn], 3], {nn, 10^8}];  (* A039701 *)
u = Map[SequencePosition[t, Flatten[Join[{ConstantArray[1, #], 2}]], 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 == 2:
            while residues.endswith("1"*n) and n <= NN:
                yield k-n
                n += 1
        residues = residues[1:] + str(resk)
print(list(islice(agen(), 13))) # Michael S. Branicky, Jan 11 2026

Crossrefs

Cf. A039701, A390511, A391796.

Sequence in context: A300772 A238244 A247816 * A356943 A017939 A282702

Adjacent sequences: A391804 A391805 A391806 * A391808 A391809 A391810

Keyword

nonn

Author

Clark Kimberling, Jan 05 2026

Extensions

a(22)-a(25) from Michael S. Branicky, Jan 11 2026

a(26)-a(31) from Jinyuan Wang, Jan 16 2026

Status

approved