A319664 Irregular triangle read by rows: T(n,k) = (-3)^k mod 2^n, n >= 2, 0 <= k <= 2^(n-2) - 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.

Links

Table of n, a(n) for n=2..64.

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

Cf. A007814, A319666.

Sequence in context: A104793 A243883 A147004 * A205961 A146620 A300291

Adjacent sequences: A319661 A319662 A319663 * A319665 A319666 A319667

Keyword

nonn,tabf

Author

Jianing Song, Sep 25 2018

Status

approved