A333851 Irregular triangle read by rows: T(n, k) = gcd(A333850(n, k), 2*A333855(n)), for n >= 1, and k = 1,2, ..., A135303(A333855(n)).

2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 2, 10, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 14, 38, 2, 2

Offset: 1

Author

Wolfdieter Lang, Jun 08 2020

Keywords

• nonn
• tabf

Comments

The length of row n is A135303(A333855(n)) (the B numbers for A333855(n)).

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:       2     2
2,   31:       1     1    1
3,   33:       1     1
4,   41:       2     2
5,   43:       1     1    1
6,   51:       2     2
7,   57:       3     3
8,   63:       2     2    2
9,   65:       2     2   10    2
10,  73:       1     1    1    1
11,  85:       2     2    2    2
12,  89:       1     1    1    1
13,  91:       2     2    2
14,  93:       2     2    2
15,  97:       2     2
16,  99:       1     1
17, 105:       6     6
18, 109:       2     2    2
19, 113:       2     2    2    2
20, 117:       2     2    2
21, 119:       2     2
22, 123:       2     2
23, 127:       1     1    1    1   1  1  1  1  1
24, 129:       1     1    1    1   1  1
25, 133:       2    14   38
26, 137:       2     2
...
		

Links

• Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Crossrefs

Cf. A333848, A333850, A333854, 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);}
pergcd(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] = gcd(vecsum(Vec(qp)), 2*n);); return (vn););}
listag(nn) = {my(v = listb(nn)); vector(#v, k, pergcd(v[k]));} \\ Michel Marcus, Jun 14 2020

Formula

T(n, k) = gcd(A333850(n, k), 2*A333855(n)), for n >= 1, and k = 1, 2, ..., A135303(A333855(n)) (B numbers >= 2 for A333855(n)).

Extensions

Some incorrect terms were found by Michel Marcus, Jun 11 2010