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A319664
Irregular triangle read by rows: T(n,k) = (-3)^k mod 2^n, n >= 2, 0 <= k <= 2^(n-2) - 1.
1
1, 1, 5, 1, 13, 9, 5, 1, 29, 9, 5, 17, 13, 25, 21, 1, 61, 9, 37, 17, 13, 25, 53, 33, 29, 41, 5, 49, 45, 57, 21, 1, 125, 9, 101, 81, 13, 89, 117, 33, 29, 41, 5, 113, 45, 121, 21, 65, 61, 73, 37, 17, 77, 25, 53, 97, 93, 105, 69, 49, 109, 57, 85
OFFSET
2,3
COMMENTS
The n-th row contains 2^(n-2) numbers, and is a permutation of 1, 5, 9, ..., 2^n - 3.
For e >= 4, the multiplicative order of a modulo 2^e equals to 2^(e-2) iff a == 3, 5 (mod 8); for e >= 5, the multiplicative order of a modulo 2^e equals to 2^(e-3) iff a == 7, 9 (mod 16); for e >= 6, the multiplicative order of a modulo 2^e equals to 2^(e-4) iff a == 15, 17 (mod 32), etc. From this we can see v(T(n,k) - 1, 2) = v(k, 2) + 2, where v(k, 2) = A007814(k) is the 2-adic valuation of k. Also, T(n,k) is a 2^v(k, 2)-th power residue but not a 2^(v(k, 2)+1)-th power residue modulo 2^i, i >= v(k, 2) + 3.
EXAMPLE
Table begins
1,
1, 5,
1, 13, 9, 5,
1, 29, 9, 5, 17, 13, 25, 21,
1, 61, 9, 37, 17, 13, 25, 53, 33, 29, 41, 5, 49, 45, 57, 21,
1, 125, 9, 101, 81, 13, 89, 117, 33, 29, 41, 5, 113, 45, 121, 21, 65, 61, 73, 37, 17, 77, 25, 53, 97, 93, 105, 69, 49, 109, 57, 85
...
PROG
(PARI) T(n, k) = lift(Mod(-3, 2^n)^k)
CROSSREFS
Sequence in context: A104793 A243883 A147004 * A205961 A146620 A300291
KEYWORD
nonn,tabf
AUTHOR
Jianing Song, Sep 25 2018
STATUS
approved