A333849 -id:A333849 - cp's OEIS frontend

A333848 a(n) gives the sum of the odd numbers of the smallest nonnegative reduced residue system modulo 2*n + 1, for n >= 0.

0, 1, 4, 9, 13, 25, 36, 32, 64, 81, 66, 121, 124, 121, 196, 225, 170, 216, 324, 240, 400, 441, 272, 529, 513, 416, 676, 560, 522, 841, 900, 570, 792, 1089, 770, 1225, 1296, 752, 1170, 1521, 1093, 1681, 1376, 1232, 1936, 1656, 1410, 1728, 2304, 1490, 2500

Offset: 0

Wolfdieter Lang, May 01 2020

The smallest nonnegative reduced residue system modulo N is the ordered set RRS(N) (written as a list) with integers k from {0, 1, ..., N-1} satisfying gcd(k, N) = 1, for N >= 1. See A038566 (with A038566(1) = 0).
If only odd members of RRS(N) are considered, name this list RRSodd(N), e.g., RRSodd(1) = [], the empty list, RRSodd(2) = [1], etc. See A216319 (but there A216319(1) = 1). The number of elements of RRSodd(N) is delta(N) = A055034(N), for N >= 2, and 0 for N = 1.
Here only numbers N = 2*n + 1 >= 1 are considered, and for the empty list RRSodd(1) a(0) is set to 0.
a(n) gives for n >= 1 also the sum of the numbers of the primitive period of the unsigned Schick sequences SBB(2*n+1, q0 = 1) (BB for Brändli and Beyne), for which 2*n + 1 satisfies A135303(n) = 1 (in Schick's notation B(2*n+1) = 1, implying initial value q0 = 1). The numbers n satisfying A135303(n) = 1 are given in A333854.
The sequence with members gcd(a(n), 2*(2*n+1)) = A333849(n) is important for a length formula for the Euler tours ET(2*n+1, q0 = 1) given in A332441(n), for n >= 1 (but A333849(n) is used only for 2*n+1 values from A333854).

			n = 4: RRSodd(9) = {1, 5, 7} with sum a(4) = 13. Schick's unsigned cycle is SBB(9, 1) = (1, 7, 5). Because A135303(4) = B(9) = 1 there is only this cycle for n = 9.

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.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Cf. A038566, A055034, A135303, A216319, A333849, A333854.

{0}~Join~Table[Total@ Select[Range[1, m, 2], GCD[#, m] == 1 &], {m, Array[2 # + 1 &, 50]}] (* Michael De Vlieger, Oct 15 2020 *)

a(n) = if (n==0, 0, my(m=2*n+1); vecsum(select(x->((gcd(m, x)==1) && (x%2)), [1..m]))); \\ Michel Marcus, May 05 2020

apply( {A333848(n)=vecsum([2*m-1|m<-[1..n],gcd(m*2-1,n*2+1)==1])}, [0..50]) \\ M. F. Hasler, Jun 04 2020

a(n) = Sum_{j=1..delta(2*n+1)} RRSodd(2*n+1)_j, for n >= 1, with delta(k) = A055034(k). a(0) = 0 (undefined case).

A332441 The lengths of the primitive periods of the partial sums of the periodic unsigned Schick sequences with initial value 1, for N = 2n + 1, for n >= 1, taken modulo 2N.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Apr 04 2020

nonn

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.

			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.

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.

Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2015-2016.

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

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

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.

More terms from Michel Marcus, Jun 14 2020

cp's OEIS Frontend

A333848 a(n) gives the sum of the odd numbers of the smallest nonnegative reduced residue system modulo 2*n + 1, for n >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A332441 The lengths of the primitive periods of the partial sums of the periodic unsigned Schick sequences with initial value 1, for N = 2n + 1, for n >= 1, taken modulo 2N.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

cp's OEIS Frontend

A333848 a(n) gives the sum of the odd numbers of the smallest nonnegative reduced residue system modulo 2*n + 1, for n >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A332441 The lengths of the primitive periods of the partial sums of the periodic unsigned Schick sequences with initial value 1, for N = 2n + 1, for n >= 1, taken modulo 2N.