A319663 - OEIS

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

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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, 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)