A206778 Irregular triangle in which n-th row lists squarefree divisors (A005117) of n.

1, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 1, 3, 1, 2, 5, 10, 1, 11, 1, 2, 3, 6, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 1, 17, 1, 2, 3, 6, 1, 19, 1, 2, 5, 10, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 6, 1, 5, 1, 2, 13, 26, 1, 3, 1, 2, 7, 14, 1, 29

Offset

1,3

Links

Reinhard Zumkeller, Rows n=1..1000 of triangle, flattened

Peter Luschny, The squarefree divisor lattice. A SageMath implementation.

Example

Triangle begins:
.   1: [1]
.   2: [1, 2]
.   3: [1, 3]
.   4: [1, 2]
.   5: [1, 5]
.   6: [1, 2, 3, 6]
.   7: [1, 7]
.   8: [1, 2]
.   9: [1, 3]
.  10: [1, 2, 5, 10]
.  11: [1, 11]
.  12: [1, 2, 3, 6].

Maple

A206778 := proc(n)
    local sqdvs, d;
    sqdvs := {} ;
    for d in numtheory[divisors](n) do
        if numtheory[issqrfree](d) then
            sqdvs := sqdvs union {d} ;
        end if;
    end do:
    sort(sqdvs) ;
end proc:
seq(op(A206778(n)), n=1..10) ; # R. J. Mathar, Mar 06 2023

Mathematica

Flatten[Table[Select[Divisors[n], SquareFreeQ], {n, 30}]] (* Harvey P. Dale, Apr 11 2012 *)

Prog

(Haskell)
a206778 n k = a206778_row n !! k
a206778_row = filter ((== 1) . a008966) . a027750_row
a206778_tabf = map a206778_row [1..]
-- Reinhard Zumkeller, May 03 2013, Feb 12 2012
(PARI) row(n) = select(x -> issquarefree(x), divisors(n)); \\ Amiram Eldar, May 02 2025
(SageMath)
def A206778_row(n: int) -> list[int]:
    pf = prime_factors(n)
    return [product(p) for k in range(len(pf) + 1) for p in Combinations(pf, k).list()]
for n in range(1, 13): print([n], A206778_row(n))  # Peter Luschny, Sep 13 2025

Crossrefs

Cf. A008966, A034444 (row lengths), A048250 (row sums), A206787; A077610.

Cf. A027750, A050320, A050326, A050328.

Sequence in context: A330569 A340785 A355758 * A101872 A251659 A174892

Adjacent sequences: A206775 A206776 A206777 * A206779 A206780 A206781

Keyword

nonn,tabf

Author

Reinhard Zumkeller, Feb 12 2012

Status

approved