

A082654


Order of 4 mod nth prime: least k such that prime(n) divides 4^k1, 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
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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^(x1) = 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 9899.
John H. Conway & R. K. Guy, The Book of Numbers, SpringerVerlag, 1996, pages 207208, 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, 240247.


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



