|
|
A222266
|
|
Irregular triangle which lists the bi-unitary divisors of n in row n.
|
|
28
|
|
|
1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 8, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 4, 7, 28, 1, 29, 1, 2, 3, 5, 6, 10, 15, 30, 1, 31, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34, 1, 5, 7, 35
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
The bi-unitary divisors of n are the divisors of n such that the largest common unitary divisor of d and n/d is 1, indicated by A165430.
The first difference from the triangle A077609 is in row n=16.
The concept of bi-unitary divisors was introduced by Suryanarayana (1972). - Amiram Eldar, Mar 09 2024
|
|
LINKS
|
|
|
EXAMPLE
|
The table starts
1;
1, 2;
1, 3;
1, 4;
1, 5;
1, 2, 3, 6;
1, 7;
1, 2, 4, 8;
1, 9;
1, 2, 5, 10;
1, 11;
1, 3, 4, 12;
1, 13;
1, 2, 7, 14;
1, 3, 5, 15;
1, 2, 8, 16;
1, 17;
|
|
MAPLE
|
# Return set of unitary divisors of n.
local u, d ;
u := {} ;
for d in numtheory[divisors](n) do
if igcd(n/d, d) = 1 then
u := u union {d} ;
end if;
end do:
u ;
end proc:
# true if d is a bi-unitary divisor of n.
isbiudiv := proc(n, d)
if n mod d = 0 then
if % = {1} then
true;
else
false;
end if;
else
false;
end if;
end proc:
# Return set of bi-unitary divisors of n
biudivs := proc(n)
local u, d ;
u := {} ;
for d in numtheory[divisors](n) do
if isbiudiv(n, d) then
u := u union {d} ;
end if;
end do:
u ;
end proc:
for n from 1 to 35 do
print(op(biudivs(n))) ;
end do:
|
|
MATHEMATICA
|
f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[Function[d, Union@ Flatten@ Select[Transpose@ {d, n/d}, Last@ Intersection[f@ #1, f@ #2] == 1 & @@ # &]]@ Select[Divisors@ n, # <= Floor@ Sqrt@ n &], {n, 35}] (* Michael De Vlieger, May 07 2017 *)
|
|
PROG
|
(PARI) isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1; }
row(n) = {my(d = divisors(n), f = factor(n), bdiv = []); for(i=1, #d, if(isbdiv(f, d[i]), bdiv = concat(bdiv, d[i]))); bdiv; } \\ Amiram Eldar, Mar 24 2023
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|