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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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).

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.

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

Table of n, a(n) for n=1..61.

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

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).

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

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

Sequence in context: A033358 A033974 A143721 * A070725 A249280 A022994

Adjacent sequences:  A333847 A333848 A333849 * A333851 A333852 A333853

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Jun 08 2020

EXTENSIONS

Some terms were corrected by Michel Marcus, Jun 11 2010

STATUS

approved

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Last modified September 21 07:28 EDT 2021. Contains 347596 sequences. (Running on oeis4.)