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A333850
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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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3
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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
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OFFSET
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1,1
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COMMENTS
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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 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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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 ...
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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
...
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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) .)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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