A114006 -id:A114006 - cp's OEIS frontend

A050328 Number of ordered factorizations of n into squarefree numbers > 1.

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, 1, 5

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000670, A000961, A002033, A002110, A005117, A008966, A050329, A114006, A206778.

import Data.List (genericIndex)
a050328 n = genericIndex a050328_list (n-1)
a050328_list = f 1 where
   f x = (if x == 1 then 1 else
         sum $ map (a050328 . (div x)) $ tail $ a206778_row x) : f (x + 1)
-- Reinhard Zumkeller, May 03 2013

a[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))); \\ Antti Karttunen, May 27 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of squarefree numbers > 1.
a(A000961(n)) = 1.
a(A002110(n)) = A000670(n).
a(n) = Sum_{d divides n, dA008966(n/d)*a(d). - Vladeta Jovovic, Sep 25 2002, corrected by Antti Karttunen, May 27 2017
G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} mu(k)^2*A(x^k). - Ilya Gutkovskiy, May 10 2019

A114004 Inverse of triangle A114002.

Original entry on oeis.org

1, -2, 1, -2, 0, 1, 2, -2, 0, 1, -2, 0, 0, 0, 1, 6, -2, -2, 0, 0, 1, -2, 0, 0, 0, 0, 0, 1, -2, 2, 0, -2, 0, 0, 0, 1, 2, 0, -2, 0, 0, 0, 0, 0, 1, 6, -2, 0, 0, -2, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -10, 6, 2, -2, 0, -2, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, -2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 6, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Nov 12 2005

Row sums are A114006. First column is A114005.

			Triangle begins
1;
-2, 1;
-2, 0, 1;
2,-2, 0, 1;
-2, 0, 0, 0, 1;
6,-2,-2, 0, 0, 1;
-2, 0, 0, 0, 0, 0, 1;
-2, 2, 0,-2, 0, 0, 0, 1;
2, 0,-2, 0, 0, 0, 0, 0, 1;
6,-2, 0, 0,-2, 0, 0, 0, 0, 1;

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

A157657 a(1) = 1, a(n) = -mu(n) for n >= 2.

Original entry on oeis.org

1, 1, 1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, -1, 0, 1, 0, 1, 0, -1, -1, 1, 0, 0, -1, 0, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, -1, -1, 0, 1, 1, 1, 0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, -1, 1, 0, 1, -1, 0, 0, -1, 1, 1, 0, -1, 1, 1, 0, 1, -1, 0, 0, -1, 1, 1, 0, 0, -1, 1, 0, -1, -1, -1, 0, 1, 0, -1, 0, -1, -1, -1, 0, 1, 0, 0, 0

Offset: 1

Jaroslav Krizek, Mar 03 2009

sign

Apparently the Dirichlet inverse of A114006. [R. J. Mathar, Jul 15 2010]

Not multiplicative; for example a(2)*a(3) != a(6). - R. J. Mathar, Mar 31 2012

Antti Karttunen, Table of n, a(n) for n = 1..5000
Index entries for sequences computed from exponents in factorization of n

Cf. A008683, A157658 (same except for a(1)).

a[1]:=1; a[n_]:=-MoebiusMu[n]; Array[a,100] (* Stefano Spezia, May 03 2025 *)

a(n)=if(n==1,1, -moebius(n)); \\ Joerg Arndt, Aug 25 2014

(define (A157657 n) (if (= 1 n) 1 (- (A008683 n)))) ;; Antti Karttunen, Jul 26 2017

Added more terms, Joerg Arndt, Aug 25 2014

cp's OEIS Frontend

A050328 Number of ordered factorizations of n into squarefree numbers > 1.

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A114004 Inverse of triangle A114002.

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A114005 First column of number triangle A114004.

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A157657 a(1) = 1, a(n) = -mu(n) for n >= 2.

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