

A319664


Irregular triangle read by rows: T(n,k) = (3)^k mod 2^n, n >= 2, 0 <= k <= 2^(n2)  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
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OFFSET

2,3


COMMENTS

The nth row contains 2^(n2) 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^(e2) iff a == 3, 5 (mod 8); for e >= 5, the multiplicative order of a modulo 2^e equals to 2^(e3) iff a == 7, 9 (mod 16); for e >= 6, the multiplicative order of a modulo 2^e equals to 2^(e4) 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 2adic 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



