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A319663
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Irregular triangle read by rows: T(n,k) = 5^k mod 2^n, n >= 2, 0 <= k <= 2^(n-2) - 1.
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1
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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
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2,3
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=2..64.
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EXAMPLE
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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
...
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PROG
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(PARI) T(n, k) = lift(Mod(5, 2^n)^k)
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CROSSREFS
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Cf. A007814, A319665.
Sequence in context: A128359 A340213 A170903 * A255166 A131113 A139426
Adjacent sequences: A319660 A319661 A319662 * A319664 A319665 A319666
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KEYWORD
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nonn,tabf
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AUTHOR
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Jianing Song, Sep 25 2018
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STATUS
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approved
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