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A082654 Order of 4 mod n-th prime: least k such that prime(n) divides 4^k-1, n >= 2. 9
0, 1, 2, 3, 5, 6, 4, 9, 11, 14, 5, 18, 10, 7, 23, 26, 29, 30, 33, 35, 9, 39, 41, 11, 24, 50, 51, 53, 18, 14, 7, 65, 34, 69, 74, 15, 26, 81, 83, 86, 89, 90, 95, 48, 98, 99, 105, 37, 113, 38, 29, 119, 12, 25, 8, 131, 134, 135, 46, 35, 47, 146, 51, 155, 78, 158 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The period of the expansion of 1/p, base N (where N=4), is equivalent to determining for base integer 4, the period of the sequence 1, 4, 4^2, 4^3, ... mod p. Thus the cycle length for base 4, 1/7 = 0.021021021... (cycle length 3).

The cycle length, base 4, mod p, is equivalent to "clock cycles", given angle A, then the algebraic identity for the doubling angle, 2A.

Examples: Given cos A, f(x) for 2A = 2x^2 - 1, seed 2 Pi/7, i.e., (.623489801 == (arrow), -.222520934... == -.900968867...== .623489801...(cycle length 3). Given 2 cos A, the algebraic identity for 2 cos 2A, f(x) = x^2 - 2; e.g., given seed 2 cos A = 2 Pi/7, the 3 cycle is 1.246979604...== .445041867...== -1.801937736...== back to 1.24697... Likewise, the doubling function given sin^2 A, f(x) for sin^2 2A = 4x(1 - x), the logistic equation; getting cycle length of 3 using the seed sin^2 2 Pi/7. Similarly, the doubling function for tan 2A given tan A, where A = 2 Pi/7 gives 2x/(1 - x^2), cycle length of 3. The doubling function for cot 2A given cot A, with A = 2 Pi/7 gives (x^2 - 1)/2x, cycle length of 3. Note that (x^2 - 1)/2x = sinh(log(x)), and is also generated from using Newton's method on x^2 + 1 = 0.

Consider the odd pseudoprimes, composite numbers x such that 2^(x-1) = 1 mod x, that have prime(n) as a factor. It appears that all such x can be factored as prime(n) * (2 a(n) k + 1) for some integer k. For example, the first few pseudoprimes having the factor 31 are 31*11, 31*91, 31*141 and 3*151. The 11th prime is 31 and a(11) = 5. Therefore all the cofactors of 31 should have the form 10k+1, which is clearly true. - T. D. Noe, Jun 10 2003

REFERENCES

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1964; Table 48, pages 98-99.

John H. Conway & R. K. Guy, The Book of Numbers, Springer-Verlag, 1996, pages 207-208, Periodic Points.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

G. Vostrov, R. Opiata, Computer modeling of dynamic processes in analytic number theory, Electrical and Computer Systems (Електротехнічні та комп'ютерні системи) 2018, No. 28, Issue 104, 240-247.

FORMULA

Least exponent k for which 4^k is congruent to 1 mod p.

EXAMPLE

4th prime is 7 and mod 7, 4^3 = 1, so a(4) = 3.

MATHEMATICA

Join[{0}, Table[MultiplicativeOrder[4, Prime[n]], {n, 2, 100}]]

PROG

(PARI) a(n)=if(n>1, znorder(Mod(4, prime(n))), 0) \\ Charles R Greathouse IV, Sep 07 2016

(GAP) A000040:=Filtered([1..350], IsPrime);;

List([1..Length(A000040)], n->OrderMod(4, A000040[n])); # Muniru A Asiru, Feb 07 2019

CROSSREFS

Cf. A014664, A002326, A036116, A036117.

Sequence in context: A194900 A194051 A195610 * A072636 A191741 A191665

Adjacent sequences:  A082651 A082652 A082653 * A082655 A082656 A082657

KEYWORD

nonn

AUTHOR

Gary W. Adamson, May 17 2003

EXTENSIONS

More terms from Reinhard Zumkeller, May 17 2003

STATUS

approved

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Last modified November 22 05:55 EST 2019. Contains 329388 sequences. (Running on oeis4.)