login
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, 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
1,1
COMMENTS
The length of row n is A135303(A333855(n)) (the B numbers for A333855(n)).
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
KEYWORD
nonn,tabf
AUTHOR
Wolfdieter Lang, Jun 08 2020
EXTENSIONS
Some incorrect terms were found by Michel Marcus, Jun 11 2010
STATUS
approved