A319663 Irregular triangle read by rows: T(n,k) = 5^k mod 2^n, n >= 2, 0 <= k <= 2^(n-2) - 1.

1, 1, 5, 1, 5, 9, 13, 1, 5, 25, 29, 17, 21, 9, 13, 1, 5, 25, 61, 49, 53, 9, 45, 33, 37, 57, 29, 17, 21, 41, 13, 1, 5, 25, 125, 113, 53, 9, 45, 97, 101, 121, 93, 81, 21, 105, 13, 65, 69, 89, 61, 49, 117, 73, 109, 33, 37, 57, 29, 17, 85, 41, 77

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, 5, 9, 13,
1, 5, 25, 29, 17, 21, 9, 13,
1, 5, 25, 61, 49, 53, 9, 45, 33, 37, 57, 29, 17, 21, 41, 13,
1, 5, 25, 125, 113, 53, 9, 45, 97, 101, 121, 93, 81, 21, 105, 13, 65, 69, 89, 61, 49, 117, 73, 109, 33, 37, 57, 29, 17, 85, 41, 77
...

Prog

(PARI) T(n, k) = lift(Mod(5, 2^n)^k)

Crossrefs

Cf. A007814, A319665.

Sequence in context: A128359 A340213 A170903 * A255166 A131113 A139426

Adjacent sequences: A319660 A319661 A319662 * A319664 A319665 A319666

Keyword

nonn,tabf

Author

Jianing Song, Sep 25 2018

Status

approved