A222266 Irregular triangle which lists the bi-unitary divisors of n in row n.

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

Offset: 1

Author

R. J. Mathar, May 05 2013

Keywords

• nonn
• tabf

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

Examples

			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;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..13171 (rows 1 <= n <= 2000).
• D. Suryanarayana, The number of bi-unitary divisors of an integer, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions, Lecture Notes in Mathematics, Vol 251, Springer, Berlin, Heidelberg, 1972.

Crossrefs

Cf. A077609, A165430, A188999 (row sums), A286324 (row lengths).

Programs

Maple

# Return set of unitary divisors of n.
A077610_row := proc(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
        A077610_row(d) intersect A077610_row(n/d) ;
        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 *)

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