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A353171
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Irregular triangle read by rows; T(n,k) = 2^k (mod prime(n)), terminating when T(n,k) = 1.
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1
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-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, -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, -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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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2,3
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COMMENTS
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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.
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).
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LINKS
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EXAMPLE
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Irregular triangle begins
n/k|| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ... || Length ||
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2 || -1 1 || 2 ||
3 || 2, -1, -2, 1 || 4 ||
4 || 2, -3, 1 || 3 ||
5 || 2, 4, -3, 5, -1, -2, -4, 3, -5, 1 || 10 ||
6 || 2, 4, -5, 3, 6, -1, -2, -4, 5, -3, -6, 1 || 12 ||
7 || 2, 4, 8, -1, -2, -4, -8, 1 || 8 ||
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PROG
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(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)
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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