A050328 -id:A050328 - cp's OEIS frontend

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

Reinhard Zumkeller, Feb 12 2012

			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].

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

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

Cf. A027750, A050320, A050326, A050328.

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

A206778 := proc(n)
    local sqdvs ,nfac,d;
    sqdvs := {} ;
    nfac := ifactors(n)[2] ;
    for d in numtheory[divisors](n) do
        if 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

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

row(n) = select(x -> issquarefree(x), divisors(n)); \\ Amiram Eldar, May 02 2025

A114006 Row sums of number triangle A114004.

Original entry on oeis.org

1, -1, -1, 1, -1, 3, -1, -1, 1, 3, -1, -5, -1, 3, 3, 1, -1, -5, -1, -5, 3, 3, -1, 7, 1, 3, -1, -5, -1, -13, -1, -1, 3, 3, 3, 13, -1, 3, 3, 7, -1, -13, -1, -5, -5, 3, -1, -9, 1, -5, 3, -5, -1, 7, 3, 7, 3, 3, -1, 31, -1, 3, -5, 1, 3, -13, -1, -5, 3, -13, -1, -25, -1, 3, -5, -5, 3, -13, -1, -9, 1, 3, -1, 31, 3, 3, 3, 7, -1, 31, 3, -5, 3, 3, 3, 11

Offset: 1

Paul Barry, Nov 12 2005

A signed version of A050328.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A008836, A008966, A050328, A114004, A114005.

a050328[n_]:=If[n==1, n, Sum[If[(dIndranil Ghosh, May 27 2017 *)

A050328(n) = if(1==n,n,sumdiv(n,d,if((dA050328(d),0)));
A114006(n) = ((-1)^bigomega(n) * A050328(n)); \\ Antti Karttunen, May 27 2017

a(n) = A008836(n) * A050328(n). - Antti Karttunen, May 27 2017

a(1) = 1; a(n) = Sum_{d|n, d < n} mu(n/d) * a(d). - Ilya Gutkovskiy, Feb 23 2020

More terms, incorrect formulas replaced by a correct one - Antti Karttunen, May 27 2017

A114005 First column of number triangle A114004.

Original entry on oeis.org

1, -2, -2, 2, -2, 6, -2, -2, 2, 6, -2, -10, -2, 6, 6, 2, -2, -10, -2, -10, 6, 6, -2, 14, 2, 6, -2, -10, -2, -26, -2, -2, 6, 6, 6, 26, -2, 6, 6, 14, -2, -26, -2, -10, -10, 6, -2, -18, 2, -10, 6, -10, -2, 14, 6, 14, 6, 6, -2, 62, -2, 6, -10, 2, 6, -26, -2, -10, 6, -26, -2, -50, -2, 6, -10, -10, 6, -26, -2, -18, 2, 6, -2, 62, 6, 6, 6, 14, -2, 62, 6, -10, 6, 6, 6

Offset: 1

Paul Barry, Nov 12 2005

Moebius transform of A114006. - Mats Granvik, Jan 02 2009

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A050328, A114006.

A050328(n) = if(1==n,n,sumdiv(n,d,if((dA050328(d),0)));
A114006(n) = ((-1)^bigomega(n) * A050328(n));
A114005(n) = if(1==n,n,2*A114006(n)); \\ Antti Karttunen, May 27 2017

a(1) = 1, for n > 1, a(n) = 2*A114006(n). - Corrected by Antti Karttunen, May 27 2017

a(1) = 1; a(n) = -2 * Sum_{d|n, d < n} a(d). - Ilya Gutkovskiy, Feb 23 2020

More terms from Antti Karttunen, May 27 2017

A050329 Number of ordered factorizations into squarefree factors indexed by prime signatures.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 7, 13, 1, 13, 9, 31, 1, 25, 11, 57, 1, 41, 101, 13, 75, 63, 91, 1, 61, 239, 15, 233, 129, 133, 1, 85, 469, 17, 535, 231, 409, 183, 1, 705, 113, 919, 321, 815, 19, 1029, 377, 1177, 241, 1, 1671, 145, 541, 2593, 681, 1301, 21, 1763, 575, 2741

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A025487, A050328.

a(n) = A050328(A025487(n)).

A385511 Numbers that are less than the number of their ordered factorizations into squarefree numbers greater than 1.

Original entry on oeis.org

2520, 5040, 7560, 10080, 10800, 12600, 15120, 20160, 21600, 22680, 23760, 25200, 27720, 30240, 32400, 35280, 37800, 43200, 45360, 47520, 50400, 52920, 55440, 60480, 64800, 65520, 70560, 71280, 75600, 79200, 83160, 86400, 88200, 90720, 95040, 98280, 100800, 105840

Offset: 1

Amiram Eldar, Jul 01 2025

nonn

Numbers k such that A050328(k) > k.
If k is a term then all the smaller numbers with the same prime signature (A118914) as k are also terms.
The least term that is not divisible by 5 is a(112) = 399168.
The least term that is not divisible by 3 is 144848704000.
The least odd term is A147516(43302) = 16639855392913235373515625.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A050328, A118914, A147516, A270308, A340155.

f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n && SquareFreeQ[n/#] &]; Select[Range[110000], f[#] > # &]

f(n) = if(n == 1, 1, sumdiv(n, d, if((d k;

A372510 Number of ordered factorizations of 2*n-1 into twin primes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 1, 2, 2, 0, 2, 1, 1, 3, 0, 1, 2, 0, 2, 2, 1, 1, 3, 2, 0, 0, 1, 1, 3, 2, 0, 1, 0, 2, 2, 0, 2, 2, 2, 0, 3, 1, 1, 6, 1, 1, 0, 0, 0, 3, 2, 1, 2, 1, 0, 2, 0, 2, 4, 1, 1, 0, 2, 2, 3, 1, 1, 3, 2, 0, 0, 0, 0, 6, 0, 1, 3, 0, 3, 2, 1

Offset: 1

Ilya Gutkovskiy, May 04 2024

nonn

			a(23) = 3 because we have 23 * 2 - 1 = 45 = 3 * 3 * 5 = 3 * 5 * 3 = 5 * 3 * 3.

Cf. A001097, A008480, A050328, A050363, A062505, A077608, A164292.

If 2*n-1 = Product A001097(k)^e(k) then a(n) = A008480(2*n-1), otherwise 0.

cp's OEIS Frontend

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

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A114006 Row sums of number triangle A114004.

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A114005 First column of number triangle A114004.

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A050329 Number of ordered factorizations into squarefree factors indexed by prime signatures.

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A385511 Numbers that are less than the number of their ordered factorizations into squarefree numbers greater than 1.

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Mathematica

PARI

A372510 Number of ordered factorizations of 2*n-1 into twin primes.

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Formula