A002321 -id:A002321 - cp's OEIS frontend

A144032 Triangle read by rows: T(n,k) = A002321(n-k+1)*A144031(k-1).

1, 0, 1, -1, 0, 1, -1, -1, 0, 0, -2, -1, -1, 0, -2, -1, -2, -1, 0, 0, -6, -2, -1, -2, 0, 2, 0, -10, -2, -2, -1, 0, 2, 6, 0, -13, -2, -2, -2, 0, 46, 10, 0, -10, -1, -2, -2, 0, 2, 12, 10, 13, 0, 4, -2, -1, -2, 0, 4, 6, 10, 13, 10, 0, 36, -2, -2, -1, 0, 4, 12, 10, 26, 10, -4, 0, 84, -3

Offset: 1

Gary W. Adamson, Sep 07 2008

Row sums = A144031, the INVERT transform of A002321.
Left border = the Mertens's function, A002321.
Right border = A144031 shifted.
Sum of n-th row terms = rightmost term of (n+1)-th row.

			First few rows of the triangle:
   1;
   0,  1;
  -1,  0,  1;
  -1, -1,  0, 0;
  -2, -1, -1, 0, -2;
  -1, -2, -1, 0,  0, -6;
  -2, -1, -2, 0,  2,  0, -10;
  -2, -2, -1, 0,  2,  6,   0, -13;
  -2, -2, -2, 0,  4,  6,  10,   0, -10;
  ...
Row 5 = (-2, -1, -1, 0, -2) termwise products of (-2, -1, -1, 0, 1) and (1, 1, 1, 0, -2); = ((-2)*(1), (-1)*(1), (-1)*(1), (0)*(0), (1)*(-2)). (-2, -1, -1, 0, 1) = the first 5 terms of A002321, the Mertens's function. (1, 1, 1, 0, -2) = 5 shifted terms of A144031.

Cf. A002321, A144031.

A162459 A002321*A000079.

Original entry on oeis.org

1, 0, -4, -8, -32, -32, -128, -256, -512, -512, -2048, -4096, -12288, -16384, -16384, -32768, -131072, -262144, -786432, -1572864, -2097152, -2097152, -8388608, -16777216, -33554432, -33554432, -67108864, -134217728, -536870912, -1610612736, -4294967296

Offset: 1

Mats Granvik, Jul 04 2009

sign

Appears to be the determinant of n X n (-1,1) matrix defined by A(i,j)=1 if j=1 or i divides j else -1.

Appears also to be the determinant of n X n (-i/j,i/j) matrix defined by A(i,j)=i/j if j=1 or i divides j else -i/j.

Cf. A002321, A000079.

from functools import lru_cache
@lru_cache(maxsize=None)
def A162459(n):
    if n == 0:
        return 0
    c, j = n, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A162459(k1)//2**(k1-1)
        j, k1 = j2, n//j2
    return 2**(n-1)*(j-c) # Chai Wah Wu, Mar 30 2021

a(n) = 2^(n-1)*A002321(n). - Chai Wah Wu, Mar 30 2021

A180050 Triangle T(n,k) read by rows. n>3,k=1 T(n,k)=A002321(n-1). The rest of the table is described by the recurrence in the Excel formula.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, -1, -2, -3, 1, -1, 1, 3, -3, 1, -2, -3, -3, 2, -3, 1, -1, 2, 2, 0, 2, -3, 1, -2, -4, -3, 0, -1, 2, -3, 1, -2, 2, 3, -3, 3, -1, 2, -3, 1, -2, -4, -4, 3, -4, 2, -1, 2, -3, 1, -1, 3, 4, -1, 0, -1, 2, -1, 2, -3, 1, -2, -5, -5, 1, 1, -1, -2, 2, -1, 2, -3, 1, -2, 3, 4, -4, 3, -1, 2

Offset: 1

Mats Granvik, Aug 08 2010

Matrix inverse of A180051. Where the Excel formula says "randbetween(-9;9)" this table has the values of the Mertens function in the first column. Help with translating the spreadsheet formula would be appreciated.

			Table begins:
1,
1,1,
1,1,1,
-1,-2,-3,1,
-1,1,3,-3,1,
-2,-3,-3,2,-3,1,
-1,2,2,0,2,-3,1,
-2,-4,-3,0,-1,2,-3,1,
-2,2,3,-3,3,-1,2,-3,1,
-2,-4,-4,3,-4,2,-1,2,-3,1,
-1,3,4,-1,0,-1,2,-1,2,-3,1,
-2,-5,-5,1,1,-1,-2,2,-1,2,-3,1,
-2,3,4,-4,3,-1,2,-2,2,-1,2,-3,1,
-3,-6,-5,3,-4,2,-2,1,-2,2,-1,2,-3,1,

Cf. A002321, A180051.

Using European dot comma style:
=if(row()>=column();if(row()<=3;1;if(column()=1; randbetween(-9;9);if(or(column()=2;column()=3);sum(indirect(address(row();column()-1; 4)))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4));sum(indirect(address(row()-column()+2; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4)))));0)

Contribution from Mats Granvik, Aug 11 2010: (Start)
[from Wouter Meeussen, seqfan]
Clear[a];
a[r_,c_]:=0 /; c>r;
a[r_,c_]:=1 /; r<=3;
a[r_,1]:=Sum[MoebiusMu[k],{k,0,r-1}];
a[r_,c_]:=a[r,c-1]-Sum[a[r-j,c], {j,1,c-1}]/; c<=3;
a[r_,c_]:=a[r,c]=Sum[a[r-j,c-1], {j,1,c-2}]-Sum[a[r-j,c], {j,1,c-1}];
(m=Table[a[i,j],{i,14},{j,14}])//ColumnForm
(End)

Contribution from Mats Granvik, Aug 11 2010: (Start)
[from Wouter Meeussen, seqfan]
a(r,c)=0 /; c>r
a(r,c)=1 /; r<=3
a(r,1)=sum(Amu(k),k=1..r)
a(r,c)=a(r,c-1)-sum(a(r-j,c), j=1..c-1)/; c<=3
a(r,c)=sum(a(r-j,c-1), j=1..c-2)-sum(a(r-j,c), j=1..c-1)
(End)

A181587 If n is even, a(n) = 0, if n is odd, a(n) = A002321(n), where A002321 is Mertens's function.

Original entry on oeis.org

1, 0, -1, 0, -2, 0, -2, 0, -2, 0, -2, 0, -3, 0, -1, 0, -2, 0, -3, 0, -2, 0, -2, 0, -2, 0, -1, 0, -2, 0, -4, 0, -3, 0, -1, 0, -2, 0, 0, 0, -1, 0, -3, 0, -3, 0, -3, 0, -3, 0, -2, 0, -3, 0, -2, 0, -1, 0, -1, 0, -2, 0, -1, 0, 0, 0, -2, 0, -1, 0, -3, 0, -4, 0, -3, 0, -2, 0, -4, 0, -4, 0, -4, 0, -3, 0, -1, 0, -2, 0, -1, 0, 0, 0, 2

Offset: 1

Mats Granvik, Oct 31 2010

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

Cf. A000035, A002321.

A181587(n) = if(n%2, sum(k=1, n, moebius(k)), 0); \\ Antti Karttunen, Sep 30 2018

a(n) = A000035(n)*A002321(n).

Definition corrected and more terms by Antti Karttunen, Sep 30 2018

Incorrect g.f. removed by Georg Fischer, Nov 09 2022

A333831 Dirichlet inverse of Mertens function (A002321).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 4, 3, 2, 5, 2, 2, 4, 3, 7, 6, 1, 2, 8, 6, 1, 6, 5, 2, 9, 4, 8, 7, 2, 9, 13, 2, 1, 6, 10, 1, 10, 3, 7, 19, 2, 3, 16, 7, 7, 6, 8, 3, 14, 10, 14, 7, 0, 1, 31, 2, 1, 19, 8, 12, 7, 2, 6, 5, 14, 3, 35, 4, 3, 23, 9, 10, 11, 4, 24

Offset: 1

Florian Lang, Apr 07 2020

Florian Lang, Table of n, a(n) for n = 1..9998

Cf. A002321.

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sum(k=1, n, moebius(k))))} \\ Andrew Howroyd, Apr 07 2020

A062983 Mertens's function of n (A002321) is not divisible by phi(n).

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Jason Earls, Jul 25 2001

nonn

Cf. A002321.

M[n_]:= Sum[MoebiusMu[k], {k, 1, n}]; Select[Range[250], Mod[M[#], EulerPhi[#]]!=0 &] (* Indranil Ghosh, Mar 16 2017 *)

M(n)=sum(k=1,n,moebius(k)); j=[]; for(n=1,250, if(Mod(M(n),eulerphi(n))!=0,j=concat(j,n))); j

A084235 Smallest k such that |M(k)| = 2^n, where M(x) is Mertens's function A002321.

Original entry on oeis.org

1, 5, 31, 199, 1637, 8507, 24110, 95961, 355541, 1786062, 6473791, 30649362, 109851909, 456774199, 2589994747, 7613644886, 37725066313, 108500046711

Offset: 1

Robert G. Wilson v, May 13 2003

Karl Sabbagh, The Riemann Hypothesis, The Greatest Unsolved Problem in Mathematics, Farrar, Straus and Giroux, New York, 2002, page 191.

Cf. A002321, A051402, A084234.

i = s = 0; Do[ While[ Abs[s] < 2^n, s = s + MoebiusMu[i]; i++ ]; Print[i - 1], {n, 0, 25}]

a(n) = A051402(2^n). - Amiram Eldar, May 06 2024

Definition corrected by L. Edson Jeffery, Mar 18 2013

a(15)-a(18) by Ashley Jordan and Luke March, Jul 22 2014

A106397 Binomial transform of Mertens's function sequence A002321.

Original entry on oeis.org

1, 1, 0, -3, -11, -30, -72, -162, -353, -756, -1598, -3334, -6868, -14002, -28368, -57374, -116233, -236165, -480781, -978293, -1984231, -4003365, -8028201, -16011129, -31812545, -63129395, -125452526, -250216779, -501567302

Offset: 0

Gary W. Adamson, May 01 2005

sign

Cf. A002321.

Binomial transform of A002321, assuming this has offset 0.

More terms from Tilman Neumann, Dec 13 2008

A319502 Indices of Mertens's function M(n) (A002321) whose nearest neighbors have value 0.

Original entry on oeis.org

165, 237, 330, 354, 357, 365, 402, 406, 421, 426, 794, 797, 813, 885, 894, 897, 905, 914, 1257, 1281, 1290, 1298, 1301, 1337, 1522, 1526, 1545, 1842, 1865, 2094, 2098, 2118, 2121, 2137, 4602, 4609, 4621, 4629, 4726, 4729, 4738, 5106, 5109, 5198, 5206, 5221

Offset: 1

Torlach Rush, Sep 20 2018

nonn

This sequence records the shortest intervals where M(n) leaves 0 before returning to 0.
a(n) - 1 and a(n) + 1 are both terms of A028442.
Both A045882 and A028442 are infinite and this allows for the possibility that this sequence is also infinite (for A028442 see comment of A002321).

			165 is a term because A002321(164) = A002321(166) = 0.
237 is a term because A002321(236) = A002321(238) = 0.

Cf. A002321, A008683, A028442, A045882.

with(numtheory): a:=n->add(mobius(k),k=1..n): select(n->a(n-1)=0 and a(n+1)=0,[$2..2200]); # Muniru A Asiru, Sep 20 2018

With[{s = Partition[Accumulate@ Array[MoebiusMu, 5300], 3, 1]}, 1 + First /@ Position[s, {0, k_, 0} /; k != 0]] (* Michael De Vlieger, Sep 24 2018 *)

isok(n) = {if (n > 1, x = sum(k=1, n-1, moebius(k)); if (x == 0, if (x + moebius(n) + moebius(n+1) == 0, return (1)););); return (0);} \\ Michel Marcus, Sep 27 2018

(A002321(a(n)) - A008683(a(n))) = (A002321(a(n)) + A008683(a(n+1))) = (A008683(a(n)) + A008683(a(n+1))) = 0.

A319520 Starts of strictly increasing runs of 0's in Mertens's function A002321.

Original entry on oeis.org

2, 39, 331, 422, 45371, 22898822, 871469945, 1319506393071, 26296710367071

Offset: 1

Torlach Rush, Sep 21 2018

a(n) is always squarefree.

It is not known whether this sequence is infinite. Sequence A045882 is infinite but it appears that increasing runs of consecutive nonsquarefree numbers thin out very quickly. The requirement that the runs consist of 0's is much stronger and makes it uncertain whether this sequence is also infinite.

			2 is a term because M(2) = 0.
39 is a term because M(39) = M(40) = 0.
331 is a term because M(331) = M(332) = M(333) = 0.
422 is a term because M(422) = ... = M(425) = 0.
45371 is a term because M(45371) = ... = M(45376) = 0.

Cf. A002321, A045882.

With[{s = Map[Boole[# == 0] &, Accumulate@ Array[MoebiusMu, 10^5]]}, Union@ Array[SequencePosition[s, ConstantArray[1, #]][[1, 1]] &, 5]] (* Michael De Vlieger, Sep 26 2018 *)

M=S=R=0;for(n=1,oo,if(!M+=moebius(n),S||S=n,S,n-S>R&&print1(S",")+R=n-S;S=0)) \\ M. F. Hasler, Nov 23 2018

a(6)-a(7) from Amiram Eldar, Sep 26 2018

a(8)-a(9) from Henri Lifchitz, Mar 09 2025

cp's OEIS Frontend

A144032 Triangle read by rows: T(n,k) = A002321(n-k+1)*A144031(k-1).

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A162459 A002321*A000079.

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A180050 Triangle T(n,k) read by rows. n>3,k=1 T(n,k)=A002321(n-1). The rest of the table is described by the recurrence in the Excel formula.

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A181587 If n is even, a(n) = 0, if n is odd, a(n) = A002321(n), where A002321 is Mertens's function.

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PARI

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A333831 Dirichlet inverse of Mertens function (A002321).

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PARI

A062983 Mertens's function of n (A002321) is not divisible by phi(n).

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Mathematica

PARI

A084235 Smallest k such that |M(k)| = 2^n, where M(x) is Mertens's function A002321.

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Mathematica

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A106397 Binomial transform of Mertens's function sequence A002321.

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A319502 Indices of Mertens's function M(n) (A002321) whose nearest neighbors have value 0.

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Maple