cp's OEIS Frontend

A353171 Irregular triangle read by rows; T(n,k) = 2^k (mod prime(n)), terminating when T(n,k) = 1.

%I A353171 #18 Apr 29 2022 17:20:27
%S A353171 -1,1,2,-1,-2,1,2,-3,1,2,4,-3,5,-1,-2,-4,3,-5,1,2,4,-5,3,6,-1,-2,-4,5,
%T A353171 -3,-6,1,2,4,8,-1,-2,-4,-8,1,2,4,8,-3,-6,7,-5,9,-1,-2,-4,-8,3,6,-7,5,
%U A353171 -9,1,2,4,8,-7,9,-5,-10,3,6,-11,1,2,4,8,-13,3,6,12,-5,-10,9,-11,7,14,-1,-2,-4,-8,13,-3,-6,-12,5,10,-9,11,-7,-14,1,2,4,8,-15,1
%N A353171 Irregular triangle read by rows; T(n,k) = 2^k (mod prime(n)), terminating when T(n,k) = 1.
%C A353171 Although the most significant digits of powers of 2 in base n are generally not periodic (the exception being when n is a power of 2), the least significant digits are. For example, 2 to an even power is congruent to 1 (mod 3) and 2 to an odd power is congruent to -1 (mod 3). This means that one can determine one of the prime factors of a Mersenne number, A000225, using the exponent. If n == 0 (mod 2), then A000225(n) == 0 (mod 3) (is a multiple of 3); if n == 0 (mod 4), then A000225(n) == 0 (mod 5); if n == 0 (mod 3), then A000225(n) == 0 (mod 7), and so on.
%C A353171 This general fact gives a reason for why certain Mersenne numbers are not prime (even with prime exponents). If p is congruent to 0 mod A014664(n) (the length of an n-th row) and prime(n) is less than the A000225(p), then prime(n) is a nontrivial factor of A000225(p).
%e A353171 Irregular triangle begins
%e A353171 n/k||  1,  2,  3,  4,  5,  6,  7,  8,  9, 10,  11, 12 ... || Length ||
%e A353171 ----------------------------------------------------------------------
%e A353171 2  || -1   1                                              ||      2 ||
%e A353171 3  ||  2, -1, -2,  1                                      ||      4 ||
%e A353171 4  ||  2, -3,  1                                          ||      3 ||
%e A353171 5  ||  2,  4, -3,  5, -1, -2, -4,  3, -5,   1             ||     10 ||
%e A353171 6  ||  2,  4, -5,  3,  6, -1, -2, -4,  5,  -3, -6,  1     ||     12 ||
%e A353171 7  ||  2,  4,  8, -1, -2, -4, -8,  1                      ||      8 ||
%o A353171 (PARI) A353171_row(n)->my(N=centerlift(Mod(2,prime(n))^1),L=List(N),k=1);while(N!=1,k++;listput(L,N=centerlift(Mod(2,prime(n))^k)));Vec(L)
%Y A353171 Cf. A000225, A014664.
%Y A353171 Cf. similar sequences: A201908, A201912.
%K A353171 sign,tabf
%O A353171 2,3
%A A353171 _Davis Smith_, Apr 28 2022