A369154 Numbers k such that A125611(k) = A125611(k + 1).

2, 9, 15, 28, 40, 41, 42, 48, 60, 68, 79, 83, 93, 95, 98, 100, 108, 114, 118, 120, 124, 129, 132, 137, 147, 149, 167, 196, 202, 206, 207, 215, 219, 221, 223, 225, 230, 243, 248, 255, 261, 265, 274, 276, 287, 299, 302, 320, 323, 329, 337, 341, 353, 356, 360, 364, 365, 373, 380, 381, 391, 405, 410

Offset: 1

Author

Robert Israel, Jan 14 2024

Keywords

• nonn

Comments

Numbers k such that A125611(k)^6 - 1 is divisible by 7^(k+1).

Since the 3 consecutive numbers 40, 41 and 42 are in the sequence, A125611(40) = A125611(41) = A125611(42) = A125611(43).

Examples

			a(3) = 15 is a term because A125611(15) = A125611(16) = 56020344873707, i.e., 56020344873707 is the least prime p such that p^6 - 1 is divisible by 7^15, and in this case p^6 - 1 is also divisible by 7^16.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..430

Crossrefs

Cf. A125611.

Programs

Maple

f:= proc(n) local R,r,i;
  R:= sort(map(rhs@op, [msolve(x^6=1, 7^n)]));
  for i from 0 do
    for r in R do
      if isprime(7^n * i + r) then return 7^n * i + r fi
  od od;
end proc:
R:= NULL: count:= 0:
for k from 1 while count < 100 do
 v:= f(k);
 if v = w then R:= R, k-1; count:= count+1 fi;
 w:= v;
od:
R;

Python

from itertools import count, islice
from sympy import nthroot_mod, isprime
def A369154_gen(): # generator of terms
    c, m = 1, 1
    for k in count(0):
        m *= 7
        r = sorted(nthroot_mod(1,6,m,all_roots=True))
        for i in count(0,m):
            for p in r:
                if isprime(i+p):
                    if i+p == c:
                        yield k
                    c = i+p
                    break
            else:
                continue
            break
A369154_list = list(islice(A369154_gen(),30)) # Chai Wah Wu, May 04 2024