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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.

%I A333850 #23 Oct 15 2020 16:36:37
%S A333850 38,26,95,71,59,103,67,224,176,175,151,115,232,184,303,219,254,170,
%T A333850 146,264,204,180,144,405,309,321,261,428,368,284,296,571,511,475,379,
%U A333850 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
%N 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.
%C A333850 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).
%C A333850 The length of row n is A135303(A333855(n)) (the B numbers for A333855(n)).
%C A333850 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.
%D A333850 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.
%H A333850 Gerold Brändli and Tim Beyne, <a href="https://arxiv.org/abs/1504.02757">Modified Congruence Modulo n with Half the Amount of Residues</a>, arXiv:1504.02757 [math.NT], 2016.
%H A333850 Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020.
%F A333850 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).
%e A333850 The irregular triangle T(n, k) begins (here A(n) = A333855(n)):
%e A333850 n,  A(n) \ k   1     2    3    4    5    6    7   8   9 ...
%e A333850 -------------------------------------------------------------
%e A333850 1,   17:      38    26
%e A333850 2,   31:      95    71   59
%e A333850 3,   33:     103    67
%e A333850 4,   41:     224   176
%e A333850 5,   43:     175   151  115
%e A333850 6,   51:     232   184
%e A333850 7,   57:     303   219
%e A333850 8,   63:     254   170  146
%e A333850 9,   65:     264   204  180  144
%e A333850 10,  73:     405   309  321  261
%e A333850 11,  85:     428   368  284  296
%e A333850 12,  89:     571   511  475  379
%e A333850 13,  91:     600   612  444
%e A333850 14,  93:     538   466  406
%e A333850 15,  97:    1254  1050
%e A333850 16,  99:     763   727
%e A333850 17, 105:     732   516
%e A333850 18, 109:     996  1080  840
%e A333850 19, 113:     952   772  688  724
%e A333850 20, 117:     844   712  556
%e A333850 21, 119:    1488  1392
%e A333850 22, 123:    1336  1144
%e A333850 23, 127:     637   517  457  469  433  385  385 361 325
%e A333850 24, 129:     649   469  469  385  397  361
%e A333850 25, 133:    1374  1218 1026
%e A333850 28, 137:    2456  2168
%e A333850 ...
%e A333850 --------------------------------------------------------------------------
%e A333850 n = 1, N = 17, B(17) = A135303((17-1)/2) = 2. In cycle notation:
%e A333850 SBB(17, q0_1) = (1, 15, 13, 9) and SBB(17, q0_2) = (3, 11, 5, 7), with sums
%e A333850 T(1, 1) = 1 + 15 + 13 + 9 = 38 and T(1, 2) = 26. (38 + 26 = 64 = A333848(8) .)
%o A333850 (PARI) RRS(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]);
%o A333850 isok8(m, n) = my(md = Mod(2, 2*n+1)^m); (md==1) || (md==-1);
%o A333850 A003558(n) = my(m=1); while(!isok8(m, n) , m++); m;
%o A333850 B(n) = eulerphi(n)/(2*A003558((n-1)/2));
%o A333850 fmiss(rrs, qs) = {for (i=1, #rrs, if (! setsearch(qs, rrs[i]), return (rrs[i])););}
%o A333850 listb(nn) = {my(v=List()); forstep (n=3, nn, 2, my(bn = B(n)); if (bn >= 2, listput(v, n););); Vec(v);}
%o A333850 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););}
%o A333850 listas(nn) = {my(v = listb(nn)); vector(#v, k, persum(v[k]));} \\ _Michel Marcus_, Jun 13 2020
%Y A333850 Cf. A332439, A333848, A333854, A333851, A333855.
%K A333850 nonn,tabf
%O A333850 1,1
%A A333850 _Wolfdieter Lang_, Jun 08 2020
%E A333850 Some terms were corrected by _Michel Marcus_, Jun 11 2010