A070549 -id:A070549 - cp's OEIS frontend

A070548 a(n) = Cardinality{ k in range 1 <= k <= n such that Moebius(k) = 1 }.

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23

Offset: 1

Benoit Cloitre, May 02 2002

Moebius(k)=1 iff k is the product of an even number of distinct primes (cf. A008683). See A057627 for Moebius(k)=0.

There was an old comment here that said a(n) was equal to A072613(n) + 1, but this is false (e.g., at n=210). - N. J. A. Sloane, Sep 10 2008

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Ed Pegg Jr., The Möbius Function (and squarefree numbers), Math Games, November 3, 2003.
Eric Weisstein's World of Mathematics, Mertens Conjecture.

Cf. A008683, A002321, A013928, A059956, A070549.

with(numtheory); M:=10000; c:=0; for n from 1 to M do if mobius(n) = 1 then c:=c+1; fi; lprint(n,c); od; # N. J. A. Sloane, Sep 14 2008

a[n_] := If[MoebiusMu[n] == 1, 1, 0]; Accumulate@ Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

for(n=1,150,print1(sum(i=1,n,if(moebius(i)-1,0,1)),","))

Asymptotics: Let N(i) = number of k in the range [1,n] with mu(k) = i, for i = 0, 1, -1. Then we know N(1) + N(-1) ~ 6n/Pi^2 (see A059956). Also, assuming the Riemann hypothesis, | N(1) - N(-1) | < n^(1/2 + epsilon) (see the Mathworld Mertens Conjecture link). Hence a(n) = N(1) ~ 3n/Pi^2 + smaller order terms. - Stefan Steinerberger, Sep 10 2008
a(n) = (1/2)*Sum_{i=1..n} (mu(i)^2 + mu(i)) = (1/2)*(A013928(n+1) + A002321(n)). - Ridouane Oudra, Oct 19 2019
From Amiram Eldar, Oct 01 2023: (Start)
a(n) = A013928(n+1) - A070549(n).
a(n) = A070549(n) + A002321(n). (End)

A081238 #{(i,j): mu(i)*mu(j) = -1, 1 <= i <= n, 1 <= j <= n}, where mu=A008683 (Moebius function).

Original entry on oeis.org

0, 2, 4, 4, 6, 12, 16, 16, 16, 24, 30, 30, 36, 48, 60, 60, 70, 70, 80, 80, 96, 112, 126, 126, 126, 144, 144, 144, 160, 176, 192, 192, 216, 240, 264, 264, 286, 312, 338, 338, 364, 390, 416, 416, 416, 448, 476, 476, 476, 476, 510, 510, 540, 540, 576, 576, 612, 648

Offset: 1

Reinhard Zumkeller, Mar 11 2003

nonn

			n  mu(n)  n: 1 2 3 4 5 6 7 8
-  -----   +----------------->
1   +1     | + - - 0 - + - 0
2   -1     | - + + 0 + - + 0
3   -1     | - + + 0 + - + 0
4    0     | 0 0 0 0 0 0 0 0
5   -1     | - + + 0 + - + 0  a(8)=16, as there are
6   +1     | + - - 0 - + - 0  16 '-1's in the 8 X 8 square
7   -1     | - + + 0 + - + 0  (represented as '-')
8    0     | 0 0 0 0 0 0 0 0

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Cf. A070548, A070549, A081239, A081240.

a081238 n = length [() | u <- [1..n], v <- [1..n],
                         a008683 u * a008683 v == -1]
-- Reinhard Zumkeller, Aug 03 2012

Nplus:= 0:
Nminus:=0:
for n from 1 to 100 do
  v:= numtheory:-mobius(n);
  if v = 1 then Nplus:= Nplus+1
  elif v = -1 then Nminus:= Nminus+1
  fi;
  A[n]:= 2*Nplus*Nminus;
od:
seq(A[n],n=1..100); # Robert Israel, Jan 08 2018

Nplus = Nminus = 0;
For[n = 1, n <= 100, n++, v = MoebiusMu[n];
     If[v == 1, Nplus++,
     If[v == -1, Nminus++]];
     a[n] = 2 Nplus Nminus];
Array[a, 100] (* Jean-François Alcover, Dec 16 2021, after Robert Israel *)

a(n) + A081239(n) + A081240(n) = n^2;
a(n) = a(n-1) iff mu(n) = 0.
a(n) = 2*A070548(n)*A070549(n). - Robert Israel, Jan 08 2018

A081240 a(n) = #{(i,j): mu(i)*mu(j) = 1, 1<=i,j<=n}, where mu = A008683 (Moebius function).

Original entry on oeis.org

1, 2, 5, 5, 10, 13, 20, 20, 20, 25, 34, 34, 45, 52, 61, 61, 74, 74, 89, 89, 100, 113, 130, 130, 130, 145, 145, 145, 164, 185, 208, 208, 225, 244, 265, 265, 290, 313, 338, 338, 365, 394, 425, 425, 425, 452, 485, 485, 485, 485, 514, 514, 549, 549, 580, 580, 613

Offset: 1

Reinhard Zumkeller, Mar 11 2003

nonn

A081238(n) + A081239(n) + a(n) = n^2;

a(n) = a(n-1) iff mu(n) = 0.

			n mu(n) ... n: 1 2 3 4 5 6 7 8
- ------ .... |---------------->
1 .. +1 ..... | + - - 0 - + - 0
2 .. -1 ..... | - + + 0 + - + 0
3 .. -1 ..... | - + + 0 + - + 0
4 ... 0 ..... | 0 0 0 0 0 0 0 0
5 .. -1 ..... | - + + 0 + - + 0 a(8)=20, as there are
6 .. +1 ..... | + - - 0 - + - 0 20 '+1's in the 8x8-square
7 .. -1 ..... | - + + 0 + - + 0 (represented as '+')
8 ... 0 ..... | 0 0 0 0 0 0 0 0.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 520 terms from Reinhard Zumkeller)

Cf. A008683, A070548, A070549.

a081240 n = length [() | u <- [1..n], v <- [1..n],
                         a008683 u * a008683 v == 1]
-- Reinhard Zumkeller, Aug 03 2012

Table[Abs[Sum[Sqrt[MoebiusMu[i]],{i,1,n}]]^2,{n,60}] (* Enrique Pérez Herrero, Jul 30 2012 *)

a(n) = |Sum_{i=1..n} sqrt(mu(i))|^2. - Enrique Pérez Herrero, Jul 30 2012

a(n) = A070548(n)^2 + A070549(n)^2. - Jason Yuen, Aug 23 2024

A346488 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(n) = 0 if mu(n) = -1, and f(n) = n for all other numbers (with mu = Möbius mu, A008683).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 2, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 2, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 38, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 2, 2, 47, 48, 2, 2, 49, 2, 50, 51, 52, 53, 2, 2, 54, 55, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71

Offset: 1

Antti Karttunen, Aug 20 2021

nonn

Restricted growth sequence transform of the sequence f(n) = 0 if mu(n) = -1, and f(n) = n for mu(n) >= 0.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j) => A305980(i) = A305980(j),
  a(i) = a(j) => b(i) = b(j), where b is the pointwise sum of any two multiplicative sequences c and d that are Dirichlet inverses of each other. For example, b can be a sequence like A319340, A323885, or A347094.

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

Cf. A008683, A070549, A030059 (positions of 2's).

Cf. also A305800, A305980, A319340, A323885, A347094.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux346488(n) = if(moebius(n)<0,0,n);
v346488 = rgs_transform(vector(up_to, n, Aux346488(n)));
A346488(n) = v346488[n];

A070549(n) = sum(k=1,n,(-1==moebius(k)));
A346488(n) = if(1==n,1,if(-1==moebius(n),2,1+n-A070549(n)));

a(1) = 1, and for n > 1, if A008683(n) = -1, a(n) = 2, otherwise a(n) = 1 + n - A070549(n).

cp's OEIS Frontend

A070548 a(n) = Cardinality{ k in range 1 <= k <= n such that Moebius(k) = 1 }.

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A081238 #{(i,j): mu(i)*mu(j) = -1, 1 <= i <= n, 1 <= j <= n}, where mu=A008683 (Moebius function).

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A081240 a(n) = #{(i,j): mu(i)*mu(j) = 1, 1<=i,j<=n}, where mu = A008683 (Moebius function).

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A346488 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(n) = 0 if mu(n) = -1, and f(n) = n for all other numbers (with mu = Möbius mu, A008683).

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Programs

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