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

Author

Wolfdieter Lang, Jun 08 2020

Keywords

• nonn
• tabf

Comments

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.

Examples

			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) .)
		

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

• 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.

Crossrefs

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

Programs

PARI

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

Formula

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).

Extensions

Some terms were corrected by Michel Marcus, Jun 11 2010