OFFSET
1,1
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).
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.
FORMULA
EXAMPLE
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) .)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Wolfdieter Lang, Jun 08 2020
EXTENSIONS
Some terms were corrected by Michel Marcus, Jun 11 2010
STATUS
approved