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A333855 Numbers 2*k + 1 with A135303(k) >= 2, for k >= 1, sorted increasingly. 6
17, 31, 33, 41, 43, 51, 57, 63, 65, 73, 85, 89, 91, 93, 97, 99, 105, 109, 113, 117, 119, 123, 127, 129, 133, 137, 145, 151, 153, 155, 157, 161, 165, 171, 177, 185, 187, 189, 193, 195, 201, 205, 209, 215, 217, 219, 221, 223, 229, 231, 233, 241, 247, 249, 251, 255 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These are the numbers a(n) for which there is more than one periodic Schick sequence. In his notation B(a(n)) >= 2, for n >= 1.

These are also the numbers a(n) for which there is more than one coach in the complete coach system Sigma(b = a(n)) of Hilton and Pedersen, for n >= 1

These are the numbers a(n) for which there is more than one cycle in the complete system MDS(a(n)) (Modified Doubling Sequence) proposed in the comment by Gary W. Adamson, Aug 20 2019, in A003558.

The complement relative to the odd numbers >= 3 is given in A333854.

The subsequence for odd primes is identical with A268923.

REFERENCES

Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, pp. 261-264.

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Z├╝rich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020

FORMULA

Sequence {a(n)}_{n>=1} of numbers 2*k + 1 satisfying A135303(k) >= 2, for k >= 1, ordered increasingly.

MATHEMATICA

1 + 2 Select[Range[2, 127], 2 <= EulerPhi[#2]/(2 If[#2 > 1 && GCD[#1, #2] == 1, Min[MultiplicativeOrder[#1, #2, {-1, 1}]], 0]) & @@ {2, 2 # + 1} &] (* Michael De Vlieger, Oct 15 2020 *)

PROG

(PARI) isok8(m, n) = my(md = Mod(2, 2*n+1)^m); (md==1) || (md==-1);

A003558(n) = my(m=1); while(!isok8(m, n) , m++); m;

isok(m) = (m%2) && eulerphi(m)/(2*A003558((m-1)/2)) >= 2; \\ Michel Marcus, Jun 09 2020

CROSSREFS

Cf. A003558, A135303, A216371, A268923, A333854.

Sequence in context: A124884 A052006 A002675 * A321217 A095748 A235920

Adjacent sequences:  A333852 A333853 A333854 * A333856 A333857 A333858

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, May 12 2020

STATUS

approved

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Last modified July 28 00:30 EDT 2021. Contains 346316 sequences. (Running on oeis4.)