A333855 -id:A333855 - cp's OEIS frontend

A333850 Irregular triangle read by rows: T(n, k) gives the sums of the members of the primitive period of the unsigned Schick sequences for the odd numbers from A333855.

38, 26, 95, 71, 59, 103, 67, 224, 176, 175, 151, 115, 232, 184, 303, 219, 254, 170, 146, 264, 204, 180, 144, 405, 309, 321, 261, 428, 368, 284, 296, 571, 511, 475, 379, 600, 612, 444, 538, 466, 406, 1254, 1050, 763, 727, 732, 516, 996, 1080, 840, 952, 772, 688, 724, 844, 712, 556, 1488, 1392, 1336, 1144

Offset: 1

Wolfdieter Lang, Jun 08 2020

For Schick's sequences see comments in A332439. In A333848 the sum for members of the primitive periods of the unsigned Schick sequences SBB(N, q0 = 1) (BB for Brändli and Beyne) for the odd numbers N from A333854 are given. (In Schick's book p is used instead of odd N >= 3, and in A333848 his B(p) = 1).
The length of row n is A135303(A333855(n)) (the B numbers for A333855(n)).
The corresponding gcd(T(n,k), 2*A333855(n)) values are given in A333851. They are used for the formula of the length of the Euler tours ET(A333855(n), q0_k), for k = 1, 2, ..., B(A333855(n)) based on the unsigned Schick sequences.

			The irregular triangle T(n, k) begins (here A(n) = A333855(n)):
n,  A(n) \ k   1     2    3    4    5    6    7   8   9 ...
-------------------------------------------------------------
1,   17:      38    26
2,   31:      95    71   59
3,   33:     103    67
4,   41:     224   176
5,   43:     175   151  115
6,   51:     232   184
7,   57:     303   219
8,   63:     254   170  146
9,   65:     264   204  180  144
10,  73:     405   309  321  261
11,  85:     428   368  284  296
12,  89:     571   511  475  379
13,  91:     600   612  444
14,  93:     538   466  406
15,  97:    1254  1050
16,  99:     763   727
17, 105:     732   516
18, 109:     996  1080  840
19, 113:     952   772  688  724
20, 117:     844   712  556
21, 119:    1488  1392
22, 123:    1336  1144
23, 127:     637   517  457  469  433  385  385 361 325
24, 129:     649   469  469  385  397  361
25, 133:    1374  1218 1026
28, 137:    2456  2168
...
--------------------------------------------------------------------------
n = 1, N = 17, B(17) = A135303((17-1)/2) = 2. In cycle notation:
SBB(17, q0_1) = (1, 15, 13, 9) and SBB(17, q0_2) = (3, 11, 5, 7), with sums
T(1, 1) = 1 + 15 + 13 + 9 = 38 and T(1, 2) = 26. (38 + 26 = 64 = A333848(8) .)

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], 2016.
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Cf. A332439, A333848, A333854, A333851, A333855.

RRS(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]);
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));
fmiss(rrs, qs) = {for (i=1, #rrs, if (! setsearch(qs, rrs[i]), return (rrs[i])););}
listb(nn) = {my(v=List()); forstep (n=3, nn, 2, my(bn = B(n)); if (bn >= 2, listput(v, n););); Vec(v);}
persum(n) = {my(bn = B(n)); if (bn >= 2, my(vn = vector(bn)); my(q=1, qt = List()); my(p = A003558((n-1)/2)); my(rrs = RRS(n)); for (k=1, bn, my(qp = List()); q = fmiss(rrs, Set(qt)); listput(qp, q); listput(qt, q); for (i=1, p-1, q = abs(n-2*q); listput(qp, q); listput(qt, q);); vn[k] = vecsum(Vec(qp));); return (vn););}
listas(nn) = {my(v = listb(nn)); vector(#v, k, persum(v[k]));} \\ Michel Marcus, Jun 13 2020

T(n, k) = Sum_{j=1..A003558(A333855(n))} SBB(A333855(n), q0_k)_j, with the unsigned Schick sequence SBB(N, q0) for all used initial values q0 = q0_k for k = 1, 2, ..., A135303(A333855(n)) (B numbers >= 2).

Some terms were corrected by Michel Marcus, Jun 11 2010

A333851 Irregular triangle read by rows: T(n, k) = gcd(A333850(n, k), 2*A333855(n)), for n >= 1, and k = 1,2, ..., A135303(A333855(n)).

Original entry on oeis.org

2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 2, 10, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 14, 38, 2, 2

Offset: 1

Wolfdieter Lang, Jun 08 2020

The length of row n is A135303(A333855(n)) (the B numbers for A333855(n)).

			The irregular triangle T(n, k) begins (here A(n) = A333855(n)):
n,  A(n) \ k   1     2    3    4   5  6  7  8  9 ...
----------------------------------------------------------------
1,   17:       2     2
2,   31:       1     1    1
3,   33:       1     1
4,   41:       2     2
5,   43:       1     1    1
6,   51:       2     2
7,   57:       3     3
8,   63:       2     2    2
9,   65:       2     2   10    2
10,  73:       1     1    1    1
11,  85:       2     2    2    2
12,  89:       1     1    1    1
13,  91:       2     2    2
14,  93:       2     2    2
15,  97:       2     2
16,  99:       1     1
17, 105:       6     6
18, 109:       2     2    2
19, 113:       2     2    2    2
20, 117:       2     2    2
21, 119:       2     2
22, 123:       2     2
23, 127:       1     1    1    1   1  1  1  1  1
24, 129:       1     1    1    1   1  1
25, 133:       2    14   38
26, 137:       2     2
...

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

Cf. A333848, A333850, A333854, A333855.

RRS(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]);
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));
fmiss(rrs, qs) = {for (i=1, #rrs, if (! setsearch(qs, rrs[i]), return (rrs[i])););}
listb(nn) = {my(v=List()); forstep (n=3, nn, 2, my(bn = B(n)); if (bn >= 2, listput(v, n););); Vec(v);}
pergcd(n) = {my(bn = B(n)); if (bn >= 2, my(vn = vector(bn)); my(q=1, qt = List()); my(p = A003558((n-1)/2)); my(rrs = RRS(n)); for (k=1, bn, my(qp = List()); q = fmiss(rrs, Set(qt)); listput(qp, q); listput(qt, q); for (i=1, p-1, q = abs(n-2*q); listput(qp, q); listput(qt, q);); vn[k] = gcd(vecsum(Vec(qp)), 2*n);); return (vn););}
listag(nn) = {my(v = listb(nn)); vector(#v, k, pergcd(v[k]));} \\ Michel Marcus, Jun 14 2020

T(n, k) = gcd(A333850(n, k), 2*A333855(n)), for n >= 1, and k = 1, 2, ..., A135303(A333855(n)) (B numbers >= 2 for A333855(n)).

Some incorrect terms were found by Michel Marcus, Jun 11 2010

A333853 The values >= 2 of A135303 for the odd numbers A333855(n), for n >= 1.

Original entry on oeis.org

2, 3, 2, 2, 3, 2, 2, 3, 4, 4, 4, 4, 3, 3, 2, 2, 2, 3, 4, 3, 2, 2, 9, 6, 3, 2, 4, 5, 2, 3, 3, 2, 2, 6, 2, 4, 2, 3, 2, 4, 2, 8, 2, 3, 6, 4, 4, 3, 3, 2, 4, 10, 3, 2, 5, 8, 16, 3, 4, 4, 6, 5, 3, 3, 4, 3, 2, 2, 2, 2

Offset: 1

Wolfdieter Lang, Jun 29 2020

nonn

In Schick's book these are the B values, the number of periodic sequences, for the odd numbers N with B values >= 2. These numbers N are given in A333855.
In the complete coach system Sigma(b) of Hilton and Pedersen, these are the number of coaches for the odd numbers b from A333855 with more than one coach.
These are also the number of periodic modified doubling sequences for the odd numbers b from A333855 given in comments and examples by Gary W. Adamson, see his Aug 25 2019 comment in A065941, where this is named "r-t table" (for roots trajectory).

			n = 23: A333855(23) = 127 with A135303((127-1)/2) = A135303(63) = 9 = a(23). There are 9 Schick cycles (see also A333850), also 9 coaches, and also 9 modified doubling sequences.

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, 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 = 1..10000
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Cf. A065941, A135303, A333850, A333855.

Map[EulerPhi[#2]/(2 If[#2 > 1 && GCD[#1, #2] == 1, Min[MultiplicativeOrder[#1, #2, {-1, 1}]], 0]) & @@ {2, #} &, 1 + 2 Select[Range[2, 15000], 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 *)

a(n) = A135303((A333855(n)-1)/2), for n >= 1.

A333854 Numbers 2*k + 1 with A135303(k) = 1, for k >= 1, sorted increasingly.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 35, 37, 39, 45, 47, 49, 53, 55, 59, 61, 67, 69, 71, 75, 77, 79, 81, 83, 87, 95, 101, 103, 107, 111, 115, 121, 125, 131, 135, 139, 141, 143, 147, 149, 159, 163, 167, 169, 173, 175, 179, 181, 183, 191, 197, 199, 203

Offset: 1

Wolfdieter Lang, May 03 2020

nonn

These are the numbers a(n) for which there is only one periodic Schick sequence. In Schick's notation B(a(n)) = 1, for n >= 1.
These are the numbers a(n) for which there is only one coach in the complete coach system Sigma(b = a(n)) of Hilton and Pedersen, for n >= 1.
These are also the numbers a(n) for which there is only 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 subsequence of prime numbers is A216371.
The complement relative to the odd numbers >= 3 is given in A333855.

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, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

Cf. A003558, A135303, A216371, A268923, A333855.

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)) == 1; \\ Michel Marcus, Jun 10 2020

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

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

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A333850 Irregular triangle read by rows: T(n, k) gives the sums of the members of the primitive period of the unsigned Schick sequences for the odd numbers from A333855.

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A333851 Irregular triangle read by rows: T(n, k) = gcd(A333850(n, k), 2*A333855(n)), for n >= 1, and k = 1,2, ..., A135303(A333855(n)).

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A333853 The values >= 2 of A135303 for the odd numbers A333855(n), for n >= 1.

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A333854 Numbers 2*k + 1 with A135303(k) = 1, for k >= 1, sorted increasingly.

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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.

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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 = 2n + 1, for n >= 1, taken modulo 2N.