A369154 - OEIS

%I #18 Jul 13 2024 17:30:58

%S 2,9,15,28,40,41,42,48,60,68,79,83,93,95,98,100,108,114,118,120,124,

%T 129,132,137,147,149,167,196,202,206,207,215,219,221,223,225,230,243,

%U 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

%N Numbers k such that A125611(k) = A125611(k + 1).

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

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

%H Robert Israel, <a href="/A369154/b369154.txt">Table of n, a(n) for n = 1..430</a>

%e 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.

%p f:= proc(n) local R,r,i;

%p   R:= sort(map(rhs@op, [msolve(x^6=1, 7^n)]));

%p   for i from 0 do

%p     for r in R do

%p       if isprime(7^n * i + r) then return 7^n * i + r fi

%p   od od;

%p end proc:

%p R:= NULL: count:= 0:

%p for k from 1 while count < 100 do

%p  v:= f(k);

%p  if v = w then R:= R, k-1; count:= count+1 fi;

%p  w:= v;

%p od:

%p R;

%o (Python)

%o from itertools import count, islice

%o from sympy import nthroot_mod, isprime

%o def A369154_gen(): # generator of terms

%o     c, m = 1, 1

%o     for k in count(0):

%o         m *= 7

%o         r = sorted(nthroot_mod(1,6,m,all_roots=True))

%o         for i in count(0,m):

%o             for p in r:

%o                 if isprime(i+p):

%o                     if i+p == c:

%o                         yield k

%o                     c = i+p

%o                     break

%o             else:

%o                 continue

%o             break

%o A369154_list = list(islice(A369154_gen(),30)) # _Chai Wah Wu_, May 04 2024

%Y Cf. A125611.

%K nonn

%O 1,1

%A _Robert Israel_, Jan 14 2024