A333851 - OEIS

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

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

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..81.

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

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: 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

...

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); }

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

CROSSREFS

Cf. A333848, A333850, A333854, A333855.

Sequence in context: A073772 A164562 A058188 * A335230 A300752 A300751

Adjacent sequences: A333848 A333849 A333850 * A333852 A333853 A333854

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Jun 08 2020

EXTENSIONS

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

STATUS

approved