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A332441 The lengths of the primitive periods of the partial sums of the periodic unsigned Schick sequences with initial value 1, for N = 2*n + 1, for n >= 1, taken modulo 2*N. 3
6, 10, 42, 54, 110, 78, 60, 68, 342, 42, 506, 250, 486, 406, 310, 330, 420, 666, 156, 410, 602, 540, 2162, 2058, 408, 1378, 220, 342, 3422, 1830, 378, 390, 4422, 1518, 4970, 1314, 1500, 2310, 6162, 4374, 6806, 680, 2436, 1958, 1092, 930, 3420, 2328, 2970, 5050, 10506 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For the signed Schick sequences see the Schick reference, where the odd N is named p. The unsigned Schick sequences are used in the Brändli and Beyne paper.

See also a comment in A332439 where the periodic unsigned Schick sequences are named SBBseq(N, q0), with B(N) = A135303((N-1)/2) different odd initial values q0 satisfying gcd(q0, N) = 1. The complete set of the primitive periods SBB(N, q0) of these sequences is named SBB(N).

The length of the primitive periods SBB(N, q0) is identical for each of the B(N) different q0 values, and named pes(N) by Schick.

Here only the lengths of the primitive periods of the partial sums of SBBseq(N, q0 = 1) (mod 2*N) is given, namely a(n) = L(2*n+1, 1).

Note that this length depends in general on the initial value q0: L(2*n+1, q0). For example, the B(65) = 4 initial values q0 = 1, 3, 7, and 11 for n = 32, N = 65, have lengths a(32) = 390, 390, 78 = 390/5, and 390, respectively.

The general length formula is L(N, q0) = 2*N*pes(N)/gcd(SUM(SBB(N, q0)), 2*N), with pes(N) = A003558((N-1)/2), and the gcd values are shown for the N values with B(N) = 1 (q0 = 1) in A333849, and for more than one initial value (B(N) >= 2) in A333851.

a(n) gives also the length of the corresponding Euler tour ET(2*n+1, q0 = 1), which may not involve all vertices of a regular (2*(2*n+1))-gon. Also the digraphs underlying these Euler tours are not always regular. See some examples below.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..51.

Gerold Brändli and Tim Beyne,Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.

FORMULA

The length a(n) = L(2*n+1 = N) = Sum_{j=1..Neff(N)} degree(Veff^{(2*N)}(j))/2, where Neff(N) is the number of vertices Veff^{(2*N)}, which are visited by the Euler tour. See the example N = 21 with Neff = 21 (not 2*N = 42) below.

a(n) = L(2*n+1 = N) = 2*N*A003558((N-1)/2)/A333849((N-1)/2), except for those N values from A333855 with the denominator replaced by the first gcd value given in the rows of array A333851. See a comment above for the general L(N, q0) formula.

EXAMPLE

n = 8 (N = 17): B(17) = 2, pes(17) = 4. SBBseq(17, 1) = repeat(1, 15, 13, 9, ),  SBBseq(17, 3) = repeat(3, 11, 5, 7, ). Euler tour ET(N, 1) = [0, 1, 16, 29, 4, 5, 20, 33, 8, 9, 24, 3, 12, 13, 28, 7, 16, 17, 32, 11, 20, 21, 2, 15, 24, 25, 6, 19, 28, 29, 10, 23, 32, 33, 14, 27, 2, 3, 18, 31, 6, 7, 22, 1, 10, 11, 26, 5, 14, 15, 30, 9, 18, 19, 0, 13, 22, 23, 4, 17, 26, 27, 8, 21, 30, 31, 12, 25, 0]. This corresponds to a regular digraph of degree 4. Neff(17) = 2*17 = 34, L(17) = 34*4/2 = 68 = a(8). Note that for N = 17 the denominator is A333851(1, 1) = 2. There is another Euler tour ET(N, 2) of the same length.

n = 10 (N = 21): B(21) = 1, pes(21) = 6. SBBseq(21, 1) = repeat(1, 19, 17, 13, 5, 11, ). The Euler tour ET(N, 1) = [0, 1, 20, 37, 8, 13, 24, 25, 2, 19, 32, 37, 6, 7, 26, 1, 14, 19, 30, 31, 8, 25, 38, 1, 12, 13, 32, 7, 20, 25, 36, 37, 14, 31, 2, 7, 18, 19, 38, 13, 26, 31, 0]. The Neff(21) = 21 vertex labels for the 42-gon are {6*k, 6*k+1, 6*k+2}, for k = 0..6. The digraph is not regular, the vertices with labels 6*k have degree 2 (visited once), for labels 6*k+1 the degree is 6, and for labels 6*k+2 the degree is 4. All other 21 vertices of the 42-gon are not involved (or have degree 0, and the connectivity number of the unconnected digraph is 22). L(21) = 7*(2/2 + 6/2 + 4/2) = 7*6 = 42 = a(10) = 2*21*6/6, because A333849(10) = 6.

PROG

(PARI) A333848(n) = if (n==0, 0, my(m=2*n+1); vecsum(select(x->((gcd(m, x)==1) && (x%2)), [1..m])));

A333849(n) = gcd(A333848(n), 2*(2*n+1));

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;

B(n) = eulerphi(n)/(2*A003558((n-1)/2));

a(n) = {my(m = 2*n+1, period = A003558(n)); if (B(m) == 1, return(2*m*period/A333849(n))); my(q=1, qs = List([q])); for (i=1, period-1, q = abs(m-2*q); listput(qs, q); ); 2*m*period/gcd(vecsum(Vec(qs)), 2*m); } \\ Michel Marcus, Jun 14 2020

CROSSREFS

Cf. A003558, A135303, A332439, A333849, A333850, A333851.

Sequence in context: A213477 A047178 A132095 * A153328 A068588 A083333

Adjacent sequences:  A332438 A332439 A332440 * A332442 A332443 A332444

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Apr 04 2020

EXTENSIONS

More terms from Michel Marcus, Jun 14 2020

STATUS

approved

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Last modified September 21 22:33 EDT 2020. Contains 337274 sequences. (Running on oeis4.)