A055615 -id:A055615 - cp's OEIS frontend

A378645 Dirichlet convolution of A055615 and A103977.

1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 3, -1, -1, -1, -1, -1, 3, -1, 1, -1, -1, -1, 3, -1, -1, -1, -1, -1, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, 5, -1, 11, -1, -1, -1, -1, -1, 3, -1, -1, -1, -1, -1, -1, -1, 7, -1, -1, -1, -3, -1, -1, -1, -1, -1, 11, -1, -1, -1, 3, -1, -1, -1, -1, -1, -1, -1, 11, -1, 5, -1, -1, -1, 3

Offset: 1

Antti Karttunen, Dec 03 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A055615, A263837, A103977, A153881, A378646 (Dirichlet inverse).

A055615(n) = (n*moebius(n));
A378645(n) = sumdiv(n,d,A055615(d)*A103977(n/d));

a(n) = Sum_{d|n} A055615(d)*A103977(n/d).

a(n) = A153881(n) always when n is a non-abundant number (A263837), and also for some of the abundant numbers, A005101.

A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 1, 4, 11, 24, 57, 112, 244, 480, 1013, 1972, 4083, 8064, 16331, 32512, 65519, 130488, 262125, 523244, 1048377, 2095104, 4194281, 8384176, 16777136, 33546240, 67108096, 134201316, 268435427, 536836584, 1073741793, 2147418112, 4294964213, 8589803520, 17179868787, 34359470272, 68719476699, 137438429184, 274877894643

Offset: 1

Antti Karttunen, Nov 22 2021

nonn

Dirichlet convolution of this sequence with phi (A000010) is A000740, with sigma (A000203) it is A034729, and with A018804 it is A034738.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A011782, A055615, A349569 (Dirichlet inverse).

Cf. also A000010, A000740, A000203, A018804, A034729, A034738, A349564, A349566, A349568.

a[n_] := DivisorSum[n, # * MoebiusMu[#] * 2^(n/# - 1) &]; Array[a, 40] (* Amiram Eldar, Nov 22 2021 *)

A055615(n) = (n*moebius(n));
A349570(n) = sumdiv(n,d,(2^(d-1)) * A055615(n/d));

a(n) = Sum_{d|n} 2^(d-1) * A055615(n/d).

A349571 Dirichlet convolution of A048673 with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 0, 1, -1, 2, -1, 4, 4, 3, -4, 4, -4, 5, 6, 13, -7, 6, -7, 5, 10, 6, -8, 10, 5, 8, 24, 9, -13, -2, -12, 40, 12, 9, 16, 16, -16, 11, 16, 11, -19, -2, -19, 8, 14, 14, -20, 28, 19, 9, 18, 12, -23, 26, 22, 21, 22, 15, -28, 2, -27, 18, 26, 121, 28, -8, -31, 11, 28, -8, -34, 46, -33, 20, 18, 15, 34, -8, -37, 29, 124

Offset: 1

Antti Karttunen, Nov 23 2021

sign

Also Dirichlet convolution of A349385 with A349387.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A048673, A055615, A349385, A349387, A349572 (Dirichlet inverse).

Cf. also A349398, A349573.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A055615(n) = (n*moebius(n));
A349571(n) = sumdiv(n,d,A048673(n/d)*A055615(d));

a(n) = Sum_{d|n} A048673(n/d) * A055615(d).

a(n) = Sum_{d|n} A349385(n/d) * A349387(d).

A144028 INVERT transform of A055615, n*mu(n).

Original entry on oeis.org

1, 1, -1, -6, -7, 3, 36, 55, -9, -221, -373, -18, 1290, 2506, 643, -7488, -16487, -7258, 42577, 106701, 65695, -236923, -681856, -534130, 1282512, 4304675, 4079414, -6687222, -26866199, -29871373, 33019148, 165771711, 212092381, -149113958, -1010995614

Offset: 0

Gary W. Adamson, Sep 07 2008

sign

Equals row sums of triangle A144029.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A055615, A144029.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*i*numtheory[mobius](i), i=1..n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 22 2017

a[n_] := a[n] = If[n == 0, 1, Sum[a[n-i]*i*MoebiusMu[i], {i, 1, n}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 18 2018, translated from Maple *)

a(4) = -7 = (0, -3, -2, 1) dot (1, 1, -1, -6); where (0, -3, -2, 1) = the first 4 terms of n*mu(n) reversed. (1, 1, -1, -6) = the first 4 terms of the INVERT recursion operation.

Corrected from a(9) onwards by R. J. Mathar, Jan 27 2011

a(0)=1 prepended by Alois P. Heinz, Sep 22 2017

A144029 Eigentriangle by rows, A055615(n-k+1)*A144028(k-1); 1<=k<=n.

Original entry on oeis.org

1, -2, 1, -3, -2, -1, 0, -3, 2, -6, -5, 0, 3, 12, -7, 6, -5, 0, 18, 14, 3, -7, 6, 5, 0, 21, -6, 36, 0, -7, -6, 30, 0, -9, -72, 55, 0, 0, 7, -36, 35, 0, -108, -110, -9, 10, 0, 0, 42, -42, -15, 0, -165, 18, -221, -11, 10, 0, 0, 49, 18, -180, 0, 27, 442, -373, 0, -11, -10, 0, 0, -21, 216, -275, 0, 663, 746, -18

Offset: 1

Gary W. Adamson, Sep 07 2008

Row sums = A144028. Right border = A144028 shifted.
Left border = A055615, n*mu(n).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
-2, 1;
-3, -2, -1;
0, -3, 2, -6;
-5, 0, 3, 12, -7;
6, -5, 0, 18, 14, 3;
-7, 6, 5, 0, 21, -6, 36;
0, -7, -6, 30, 0, -9, -72, 55;
0, 0, 7, -36, 35, 0, -108, -110, -9;
10, 0, 0, 42, -42, -15, 0, -165, 18, -221;
...
Row 4 = (0, -3, 2, -6) = termwise products of (0, -3, -2, 1) and (1, 1, -1, -6) = (0*1, -3*1, -2*-1, 1*(-6)). (0, -3, -2, 1) = the first 4 terms of A055615, n*mu(n), reversed.
(1, 1, -1, 6) = the first 4 terms A144028, shifted.

Cf. A055615, A144028.

read("transforms");
A055615 := proc(n) n*numtheory[mobius](n) ; end proc:
A144028 :=proc(n) if n = 0 then 1; else L := [seq(A055615(i),i=1..n+2)] ; INVERT(L) ; op(n,%) ; end if; end proc:
A144029 := proc(n,k) A055615(n-k+1)*A144028(k-1) ; end proc: # R. J. Mathar, Jan 27 2011

Eigentriangle by rows, A055615(n-k+1)*A144028(k-1); 1<=k<=n.

Entries corrected starting from row 10. - R. J. Mathar, Jan 27 2011

A383286 Dirichlet convolution of A276086 (primorial base exp-function) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

2, -1, 0, 3, 8, -4, -4, -3, 12, 14, 68, 6, 24, 62, 96, 195, 416, 86, 212, 270, 720, 956, 2204, 584, 1160, 1788, 3660, 5454, 11192, -300, -48, -429, -228, -820, 20, -260, -4, -376, 60, -420, 548, -1462, 264, -1758, 540, -2902, 3056, -960, 1680, 80, 3900, 4086, 15644, -3320, 8212, 1896, 25500, 16904, 78632, -850, -24, 150

Offset: 1

Antti Karttunen, May 12 2025

sign

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base.

Cf. A055615, A276086.

Cf. also A349394, A369010.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A055615(n) = (n*moebius(n));
A383286(n) = sumdiv(n,d,A276086(n/d)*A055615(d));

a(n) = Sum_{d|n} A276086(n/d)*A055615(d).

A369455 Dirichlet convolution of A083345 with A055615 (Dirichlet inverse of n), where A083345(n) = (n'/gcd(n,n')) and n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, -1, 1, 0, 1, 1, -1, 0, 1, -3, 1, 0, 0, -4, 1, -6, 1, -3, 0, 0, 1, 0, -3, 0, -5, -3, 1, 0, 1, 1, 0, 0, 0, 9, 1, 0, 0, 0, 1, 0, 1, -3, -6, 0, 1, -3, -5, -20, 0, -3, 1, -8, 0, 0, 0, 0, 1, 0, 1, 0, -6, -7, 0, 0, 1, -3, 0, 0, 1, -6, 1, 0, -20, -3, 0, 0, 1, -3, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0, 1, -42, -6, 29

Offset: 1

Antti Karttunen, Jan 27 2024

sign

Dirichlet convolution of this sequence with A000010 (Euler phi) is A369068.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A055615, A083345.

Cf. also A349394, A349396, A369068, A369069.

A055615(n) = (n*moebius(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369455(n) = sumdiv(n, d, A083345(n/d)*A055615(d));

a(n) = Sum_{d|n} A055615(n/d)*A083345(d).

A144966 Square array T(n,k) read by antidiagonals up. A055615 interleaved with k-1 zeros in each column. Redheffer type matrix.

Original entry on oeis.org

1, -2, 1, -3, 1, 1, 0, 0, 0, 1, -5, -2, 1, 0, 1, 6, 0, 0, 0, 0, 1, -7, -3, 0, 1, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, -11, -5, -3, -2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -13, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 14, 0, 0, 0, -2, 0

Offset: 1

Mats Granvik, Sep 27 2008

Determinant of this array appears to be the triangular numbers. The number of permutations that contribute to the result of the zero corner determinant that give the natural numbers, appears to be given by A002033.

			Determinant of:
1
is equal to 1.
Determinant of:
1,1
-2,0
is equal to 2.
Determinant of:
1,1,1
-2,1,0
-3,0,0
is equal to 3.
Determinant of:
1,1,1,1
-2,1,0,0
-3,0,1,0
0,-2,0,0
is equal to 4.
Determinant of:
1,1,1,1,1
-2,1,0,0,0
-3,0,1,0,0
0,-2,0,1,0
-5,0,0,0,0
is equal to 5.
Determinant of:
1,1,1,1,1,1
-2,1,0,0,0,0
-3,0,1,0,0,0
0,-2,0,1,0,0
-5,0,0,0,1,0
6,-3,-2,0,0,0
is equal to 6.

Cf. A143104, A143142.

=if(mod(row(); column())=0; lookup(row()/column(); A000027; A055615); if(row()=1; 1; 0))

A008683 Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.

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

Offset: 1

N. J. A. Sloane

Moebius inversion: f(n) = Sum_{d|n} g(d) for all n <=> g(n) = Sum_{d|n} mu(d)*f(n/d) for all n.
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).
A008683 = A140579^(-1) * A140664. - Gary W. Adamson, May 20 2008
Coons & Borwein prove that Sum_{n>=1} mu(n) z^n is transcendental. - Jonathan Vos Post, Jun 11 2008; edited by Charles R Greathouse IV, Sep 06 2017
Equals row sums of triangle A144735 (the square of triangle A054533). - Gary W. Adamson, Sep 20 2008
Conjecture: a(n) is the determinant of Redheffer matrix A143104 where T(n, n) = 0. Verified for the first 50 terms. - Mats Granvik, Jul 25 2008
From Mats Granvik, Dec 06 2008: (Start)
The Editorial Office of the Journal of Number Theory kindly provided (via B. Conrey) the following proof of the conjecture: Let A be A143104 and B be A143104 where T(n, n) = 0.
"Suppose you expand det(B_n) along the bottom row. There is only a 1 in the first position and so the answer is (-1)^n times det(C_{n-1}) say, where C_{n-1} is the (n-1) by (n-1) matrix obtained from B_n by deleting the first column and the last row. Now the determinant of the Redheffer matrix is det(A_n) = M(n) where M(n) is the sum of mu(m) for 1 <= m <= n. Expanding det(A_n) along the bottom row, we see that det(A_n) = (-1)^n * det(C_{n-1}) + M(n-1). So we have det(B_n) = (-1)^n * det(C_{n-1}) = det(A_n) - M(n-1) = M(n) - M(n-1) = mu(n)." (End)
Conjecture: Consider the table A051731 and treat 1 as a divisor. Move the value in the lower right corner vertically to a divisor position in the transpose of the table and you will find that the determinant is the Moebius function. The number of permutation matrices that contribute to the Moebius function appears to be A074206. - Mats Granvik, Dec 08 2008
Convolved with A152902 = A000027, the natural numbers. - Gary W. Adamson, Dec 14 2008
[Pickover, p. 226]: "The probability that a number falls in the -1 mailbox turns out to be 3/Pi^2 - the same probability as for falling in the +1 mailbox". - Gary W. Adamson, Aug 13 2009
Let A = A176890 and B = A * A * ... * A, then the leftmost column in matrix B converges to the Moebius function. - Mats Granvik, Gary W. Adamson, Apr 28 2010 and May 28 2020
Equals row sums of triangle A176918. - Gary W. Adamson, Apr 29 2010
Calculate matrix powers: A175992^0 - A175992^1 + A175992^2 - A175992^3 + A175992^4 - ... Then the Mobius function is found in the first column. Compare this to the binomial series for (1+x)^-1 = 1 - x + x^2 - x^3 + x^4 - ... . - Mats Granvik, Gary W. Adamson, Dec 06 2010
From Richard L. Ollerton, May 08 2021: (Start)
Formulas for the numerous OEIS entries involving the Möbius transform (Dirichlet convolution of a(n) and some sequence h(n)) can be derived using the following (n >= 1):
Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.
Use of gcd(n,k)*lcm(n,k) = n*k provides further variations. (End)
Formulas for products corresponding to the sums above are also available for sequences f(n) > 0: Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))). - Richard L. Ollerton, Nov 08 2021

			G.f. = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 64-65.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
Clifford A. Pickover, "The Math Book, from Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics", Sterling Publishing, 2009, p. 226. - Gary W. Adamson, Aug 13 2009
G. Pólya and G. Szegő, Problems and Theorems in Analysis Volume II. Springer_Verlag 1976.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 98-99.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 826.
All Angles, What is the Moebius function?, video, 2024.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 705-707.
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Olivier Bordellès, Some Explicit Estimates for the Mobius Function , J. Int. Seq. 18 (2015) 15.11.1
G. J. Chaitin, Thoughts on the Riemann hypothesis arXiv:math/0306042 [math.HO], 2003.
Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, arXiv:0806.1563 [math.NT], 2008.
Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295.
Tom Edgar, Posets and Möbius Inversion, slides, (2008).
Mats Granvik, Inverse of a triangular matrix using determinants, Inverse of a triangular matrix using matrix multiplication, Inverse of a triangular matrix as a binomial series, The ordinary generating function for the Mobius function.
Keith Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n).
A. F. Möbius, Über eine besondere Art von Umkehrung der Reihen. Journal für die reine und angewandte Mathematik 9 (1832), 105-123.
Ed Pegg Jr., The Mobius function (and squarefree numbers).
Anders Björner and Richard P. Stanley, A combinatorial miscellany.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects Of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238.
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, Jun 05 2014, pp. 105-124.
Gerard Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens.
Eric Weisstein's World of Mathematics, Moebius Function.
Eric Weisstein's World of Mathematics, Redheffer Matrix.
Wikipedia, Moebius function.
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Variants of a(n) are A178536, A181434, A181435.
Cf. A000010, A001221, A008966, A007423, A080847, A002321 (partial sums), A069158, A055615, A129360, A140579, A140664, A140254, A143104, A152902, A206706, A063524, A007427, A007428, A124010, A073776, A074206, A132971, A156552.
Cf. A059956 (Dgf at s=2), A088453 (Dgf at s=3), A215267 (Dgf at s=4), A343308 (Dgf at s=5).

Axiom
```
[moebiusMu(n) for n in 1..100]
```

import Math.NumberTheory.Primes.Factorisation (factorise)
a008683 = mu . snd . unzip . factorise where
mu [] = 1; mu (1:es) = - mu es; mu (_:es) = 0
-- Reinhard Zumkeller, Dec 13 2015, Oct 09 2013

a008683 1 = 1
a008683 n = - sum [a008683 d | d <- [1..(n-1)], n `mod` d == 0]
-- Harry Richman, Jun 13 2025

Magma
```
[ MoebiusMu(n) : n in [1..100]];
```

with(numtheory): A008683 := n->mobius(n);
with(numtheory): [ seq(mobius(n), n=1..100) ];
# Note that older versions of Maple define mobius(0) to be -1.
# This is unwise! Moebius(0) is better left undefined.
with(numtheory):
mu:= proc(n::posint) option remember; `if`(n=1, 1,
       -add(mu(d), d=divisors(n) minus {n}))
     end:
seq(mu(n), n=1..100);  # Alois P. Heinz, Aug 13 2008

Array[ MoebiusMu, 100]
(* Second program: *)
m = 100; A[_] = 0;
Do[A[x_] = x - Sum[A[x^k], {k, 2, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Oct 20 2019, after Ilya Gutkovskiy *)

A008683(n):=moebius(n)$ makelist(A008683(n),n,1,30); /* Martin Ettl, Oct 24 2012 */

PARI
```
a=n->if(n<1,0,moebius(n));
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 - X)[n])};

list(n)=my(v=vector(n,i,1)); forprime(p=2, sqrtint(n), forstep(i=p, n, p, v[i]*=-1); forstep(i=p^2, n, p^2, v[i]=0)); forprime(p=sqrtint(n)+1, n, forstep(i=p, n, p, v[i]*=-1)); v \\ Charles R Greathouse IV, Apr 27 2012

from sympy import mobius
print([mobius(i) for i in range(1, 101)])  # Indranil Ghosh, Mar 18 2017

@cached_function
def mu(n):
    if n < 2: return n
    return -sum(mu(d) for d in divisors(n)[:-1])
# Changing the sign of the sum gives the number of ordered factorizations of n A074206.
print([mu(n) for n in (1..96)])  # Peter Luschny, Dec 26 2016

Sum_{d|n} mu(d) = 1 if n = 1 else 0.
Dirichlet generating function: Sum_{n >= 1} mu(n)/n^s = 1/zeta(s). Also Sum_{n >= 1} mu(n)*x^n/(1-x^n) = x.
In particular, Sum_{n > 0} mu(n)/n = 0. - Franklin T. Adams-Watters, Jun 20 2014
phi(n) = Sum_{d|n} mu(d)*n/d.
a(n) = A091219(A091202(n)).
Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1. - David W. Wilson, Aug 01 2001
abs(a(n)) = Sum_{d|n} 2^A001221(d)*a(n/d). - Benoit Cloitre, Apr 05 2002
Sum_{d|n} (-1)^(n/d)*mobius(d) = 0 for n > 2. - Emeric Deutsch, Jan 28 2005
a(n) = (-1)^omega(n) * 0^(bigomega(n) - omega(n)) for n > 0, where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Dirichlet generating function for the absolute value: zeta(s)/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
mu(n) = A129360(n) * (1, -1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007
mu(n) = -Sum_{d < n, d|n} mu(d) if n > 1 and mu(1) = 1. - Alois P. Heinz, Aug 13 2008
a(n) = A174725(n) - A174726(n). - Mats Granvik, Mar 28 2010
a(n) = first column in the matrix inverse of a triangular table with the definition: T(1, 1) = 1, n > 1: T(n, 1) is any number or sequence, k = 2: T(n, 2) = T(n, k-1) - T(n-1, k), k > 2 and n >= k: T(n,k) = (Sum_{i = 1..k-1} T(n-i, k-1)) - (Sum_{i = 1..k-1} T(n-i, k)). - Mats Granvik, Jun 12 2010
Product_{n >= 1} (1-x^n)^(-a(n)/n) = exp(x) (product form of the exponential function). - Joerg Arndt, May 13 2011
a(n) = Sum_{k=1..n, gcd(k,n)=1} exp(2*Pi*i*k/n), the sum over the primitive n-th roots of unity. See the Apostol reference, p. 48, Exercise 14 (b). - Wolfdieter Lang, Jun 13 2011
mu(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n. (conjecture). - Mats Granvik, Nov 20 2011
Sum_{k=1..n} a(k)*floor(n/k) = 1 for n >= 1. - Peter Luschny, Feb 10 2012
a(n) = floor(omega(n)/bigomega(n))*(-1)^omega(n) = floor(A001221(n)/A001222(n))*(-1)^A001221(n). - Enrique Pérez Herrero, Apr 27 2012
Multiplicative with a(p^e) = binomial(1, e) * (-1)^e. - Enrique Pérez Herrero, Jan 19 2013
G.f. A(x) satisfies: x^2/A(x) = Sum_{n>=1} A( x^(2*n)/A(x)^n ). - Paul D. Hanna, Apr 19 2016
a(n) = -A008966(n)*A008836(n)/(-1)^A005361(n) = -floor(rad(n)/n)Lambda(n)/(-1)^tau(n/rad(n)). - Anthony Browne, May 17 2016
a(n) = Kronecker delta of A001221(n) and A001222(n) (which is A008966) multiplied by A008836(n). - Eric Desbiaux, Mar 15 2017
a(n) = A132971(A156552(n)). - Antti Karttunen, May 30 2017
Conjecture: a(n) = Sum_{k>=0} (-1)^(k-1)*binomial(A001222(n)-1, k)*binomial(A001221(n)-1+k, k), for n > 1. Verified for the first 100000 terms. - Mats Granvik, Sep 08 2018
From Peter Bala, Mar 15 2019: (Start)
Sum_{n >= 1} mu(n)*x^n/(1 + x^n) = x - 2*x^2. See, for example, Pólya and Szegő, Part V111, Chap. 1, No. 71.
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 - x^n) = x + 2*(x^2 + x^4 + x^8 + x^16 + ...).
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 + x^n) = x - 2*(x^4 + x^8 + x^16 + x^32 + ...).
Sum_{n >= 1} |mu(n)|*x^n/(1 - x^n) = Sum_{n >= 1} (2^w(n))*x^n, where w(n) is the number of different prime factors of n (Hardy and Wright, Chapter XVI, Theorem 264).
Sum_{n odd} |mu(n)|*x^n/(1 + x^(2*n)) = Sum_{n in S_1} (2^w_1(n))*x^n, where S_1 = {1, 5, 13, 17, 25, 29, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 1 (mod 4), and w_1(n) is the number of different prime factors p = 1 (mod 4) of n.
Sum_{n odd} (-1)^((n-1)/2)*mu(n)*x^n/(1 - x^(2*n)) = Sum_{n in S_3} (2^w_3(n))*x^n, where S_3 = {1, 3, 7, 9, 11, 19, 21, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 3 (mod 4), and where w_3(n) is the number of different prime factors p = 3 (mod 4) of n. (End)
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, May 11 2019
a(n) = sign(A023900(n)) * [A007947(n) = n] where [] is the Iverson bracket. - I. V. Serov, May 15 2019
a(n) = Sum_{k = 1..n} gcd(k, n)*a(gcd(k, n)) = Sum_{d divides n} a(d)*d*phi(n/d). - Peter Bala, Jan 16 2024

A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 2, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77

Offset: 1

R. Muller, Mar 15 1996

Also called core(n). [Not to be confused with the squarefree kernel of n, A007947.]
Sequence read mod 4 gives A065882. - Philippe Deléham, Mar 28 2004
This is an arithmetic function and is undefined if n <= 0.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), lcm(A007947(b),c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n. [Corrected by M. F. Hasler, Mar 01 2018]
If n > 1, the quantity f(n) = log(n/core(n))/log(n) satisfies 0 <= f(n) <= 1; f(n) = 0 when n is squarefree and f(n) = 1 when n is a perfect square. One can define n as being "epsilon-almost squarefree" if f(n) < epsilon. - Kurt Foster (drsardonicus(AT)earthlink.net), Jun 28 2008
a(n) is the smallest natural number m such that product of geometric mean of the divisors of n and geometric mean of the divisors of m are integers. Geometric mean of the divisors of number n is real number b(n) = Sqrt(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A000290: a(A000290(n)) = 1. For n = 8; b(8) = sqrt(8), a(n) = 2 because b(2) = sqrt(2); sqrt(8) * sqrt(2) = 4 (integer). - Jaroslav Krizek, Apr 26 2010
Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011
Booker, Hiary, & Keating outline a method for bounding (on the GRH) a(n) for large n using L-functions. - Charles R Greathouse IV, Feb 01 2013
According to the formula a(n) = n/A000188(n)^2, the scatterplot exhibits the straight lines y=x, y=x/4, y=x/9, ..., i.e., y=x/k^2 for all k=1,2,3,... - M. F. Hasler, May 08 2014
The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - Álvar Ibeas, Mar 19 2015
a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - Zak Seidov, Jan 30 2016
All solutions of the Diophantine equation n*x=y^2 or, equivalently, G(n,x)=y, with G being the geometric mean, are of the form x=k^2*a(n), y=k*sqrt(n*a(n)), where k is a positive integer. - Stanislav Sykora, Feb 03 2016
If f is a multiplicative function then Sum_{d divides n} f(a(d)) is also multiplicative. For example, A010052(n) = Sum_{d divides n} mu(a(d)) and A046951(n) = Sum_{d divides n} mu(a(d)^2). - Peter Bala, Jan 24 2024

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Krassimir T. Atanassov, On the 22nd, 23rd, and the 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.
Andrew Booker, Ghaith Hiary, and Jon Keating, Detecting squarefree numbers, CNTA XII (2012).
Henry Bottomley, Some Smarandache-type multiplicative sequences.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Vlad Copil and Laurenţiu Panaitopol, Properties of a sequence generated by positive integers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 2 (2007), pp. 131-137; alternative link.
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Eric Weisstein's World of Mathematics, Squarefree Part.

See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.
See A027746, A027748, A124010 for factorization data for n.
Analogous sequences: A050985, A053165, A055231.
Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums), A090699, A350389.
Related to A006519 via A225546.

a007913 n = product $
            zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)
-- Reinhard Zumkeller, Jul 06 2012

[ Squarefree(n) : n in [1..256] ]; // N. J. A. Sloane, Dec 23 2006

A007913 := proc(n) local f,a,d; f := ifactors(n)[2] ; a := 1 ; for d in f do if type(op(2,d),'odd') then a := a*op(1,d) ; end if; end do: a; end proc: # R. J. Mathar, Mar 18 2011
# second Maple program:
a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 20 2015
seq(n / expand(numtheory:-nthpow(n, 2)), n=1..77);  # Peter Luschny, Jul 12 2022

data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[] -> 1; sfp = data/sp /. Sqrt[x] -> x (* Artur Jasinski, Nov 03 2008 *)
Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,100}] (* Zak Seidov, Apr 08 2009 *)
Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* T. D. Noe, May 20 2013 *)
Sqrt[#] /. (c_:1)*a_^(b_:0) -> (c*a^b)^2& /@ Range@100 (* Bill Gosper, Jul 18 2015 *)

PARI
```
a(n)=core(n)
```

from sympy import factorint, prod
def A007913(n):
    return prod(p for p, e in factorint(n).items() if e % 2)
# Chai Wah Wu, Feb 03 2015

[squarefree_part(n) for n in (1..77)] # Peter Luschny, Feb 04 2015

Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson, Aug 01 2001
a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - Philippe Deléham, Mar 28 2004
Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - Anthony Browne, Jun 06 2016
a(n) = rad(n)/a(n/rad(n)), where rad = A007947. This recurrence relation together with a(1) = 1 generate the sequence. - Velin Yanev, Sep 19 2017
From Peter Munn, Nov 18 2019: (Start)
a(k*m) = A059897(a(k), a(m)).
a(n) = n / A008833(n).
(End)
a(A225546(n)) = A225546(A006519(n)). - Peter Munn, Jan 04 2020
From Amiram Eldar, Mar 14 2021: (Start)
Theorems proven by Copil and Panaitopol (2007):
Lim sup_{n->oo} a(n+1)-a(n) = oo.
Lim inf_{n->oo} a(n+1)-a(n) = -oo.
Sum_{k=1..n} 1/a(k) ~ c*sqrt(n) + O(log(n)), where c = zeta(3/2)/zeta(3) (A090699). (End)
a(n) = A019554(n)^2/n. - Jianing Song, May 08 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Oct 25 2022
a(n) = A007947(A350389(n)). - Amiram Eldar, Jan 20 2024

More terms from Michael Somos, Nov 24 2001

Definition reformulated by Daniel Forgues, Mar 24 2009

cp's OEIS Frontend

A378645 Dirichlet convolution of A055615 and A103977.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349571 Dirichlet convolution of A048673 with A055615 (Dirichlet inverse of n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A144028 INVERT transform of A055615, n*mu(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A144029 Eigentriangle by rows, A055615(n-k+1)*A144028(k-1); 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A383286 Dirichlet convolution of A276086 (primorial base exp-function) with A055615 (Dirichlet inverse of n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A369455 Dirichlet convolution of A083345 with A055615 (Dirichlet inverse of n), where A083345(n) = (n'/gcd(n,n')) and n' is the arithmetic derivative of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A144966 Square array T(n,k) read by antidiagonals up. A055615 interleaved with k-1 zeros in each column. Redheffer type matrix.

Views

Author

Keywords

Comments

Examples