A261890 -id:A261890 - cp's OEIS frontend

A055615 a(n) = n * mu(n), where mu is the Möbius function A008683.

1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, -13, 14, 15, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -30, -31, 0, 33, 34, 35, 0, -37, 38, 39, 0, -41, -42, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -66, -67, 0, 69, -70, -71, 0

Offset: 1

Michael Somos, Jun 04 2000

Dirichlet inverse of n (A000027).
Absolute values give n if n is squarefree, otherwise 0.
a(n) is multiplicative because both mu(n) and n are. - Mitch Harris, Jun 09 2005
a(n) is multiplicative with a(p^1) = -p, a(p^e) = 0 if e > 1. - David W. Wilson, Jun 12 2005
Negative of the Moebius number of the dihedral group of order 2n. - Eric M. Schmidt, Jul 28 2013

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

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Mats Granvik, Primes approximated by eigenvalues.
Mats Granvik, Mobius function times n approximated by eigenvalues.

Moebius transform of A023900.
Cf. A000027 (Dirichlet inverse), A061669 (sum with it).
Cf. A062004.
Cf. A013929 (positions of 0's), A068340 (partial sums), A261869 (first differences), A261890 (second differences).
Cf. A008683, A334657, A334659, A334660.

a055615 n = a008683 n * n  -- Reinhard Zumkeller, Sep 04 2015

[n*MoebiusMu(n): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014

with(numtheory): A055615:=n->n*mobius(n): seq(A055615(n), n=1..100); # Wesley Ivan Hurt, Nov 18 2014

Table[n MoebiusMu[n], {n,80}] (* Harvey P. Dale, May 26 2011 *)

PARI
```
{a(n) = if( n<1, 0, n * moebius(n))};
```

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

from sympy import mobius
def A055615(n): return n*mobius(n) # Chai Wah Wu, Apr 01 2023

[n*moebius(n) for n in (1..100)] # G. C. Greubel, May 24 2022

a(n) = n * A008683(n).
Dirichlet g.f.: 1/zeta(s-1).
Multiplicative with a(p^e) = -p*0^(e-1), e>0 and p prime. - Reinhard Zumkeller, Jul 17 2003
Conjectures: lim b->1+ Sum n=1..inf a(n)*b^(-n) = -12 and lim b->1- Sum n=1..inf a(n)*b^n = -12 (+ indicates that b decreases to 1, - indicates it increases to 1), both considering that zeta(-1) = -1/12 and calculations (more generally mu(n)*n^s is Abel summable to zeta(-s)). - Gerald McGarvey, Sep 26 2004
Dirichlet generating function for the absolute value: zeta(s-1)/zeta(2s-2). - Franklin T. Adams-Watters, Sep 11 2005
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k*A(x^k). - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ 3*n^2/Pi^2. - Amiram Eldar, Feb 02 2024

A261869 First differences of A055615.

Original entry on oeis.org

-3, -1, 3, -5, 11, -13, 7, 0, 10, -21, 11, -13, 27, 1, -15, -17, 17, -19, 19, 21, 1, -45, 23, 0, 26, -26, 0, -29, -1, -1, 31, 33, 1, 1, -35, -37, 75, 1, -39, -41, -1, -1, 43, 0, 46, -93, 47, 0, 0, 51, -51, -53, 53, 55, -55, 57, 1, -117, 59, -61, 123, -62, 0

Offset: 1

Reinhard Zumkeller, Sep 04 2015

sign

a(n) = (n+1)*mu(n+1) - n*mu(n), where mu = A008683 (Moebius function);
a(A068781(n)) = 0;
abs(a(n)) <= 2*n + 1.

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

Cf. A008683, A055615, A068781, A261890 (first differences).

import Data.List (genericIndex)
a261869 n = genericIndex a261869_list (n - 1)
a261869_list = zipWith (-) (tail a055615_list) a055615_list

#[[2]]-#[[1]]&/@Partition[Table[n*MoebiusMu[n],{n,70}],2,1] (* Harvey P. Dale, Apr 17 2016 *)

A114180 Numbers n with mu(n) = mu(n+1) = mu(n+2).

Original entry on oeis.org

29, 33, 41, 48, 85, 93, 98, 101, 124, 137, 141, 201, 213, 217, 229, 242, 243, 281, 301, 342, 350, 393, 423, 429, 433, 445, 475, 548, 603, 617, 633, 641, 645, 697, 724, 741, 774, 821, 844, 845, 846, 869, 921, 969, 1021, 1024, 1041, 1085, 1129, 1137, 1189

Offset: 1

Franklin T. Adams-Watters, Feb 03 2006

nonn

Any sequence of 4 or more consecutive numbers with the same value for mu must all have mu(n)=0 (n divisible by a proper square) since at least one of every 4 consecutive numbers is divisible by 4.

A261890(a(n)) = 0. - Reinhard Zumkeller, Sep 05 2015

			mu(n)=1 for 33,34,35; 85,86,87; 93,94,95; ...
mu(n)=-1 for 29,30,31; 41,42,43; 101,102,103; ...
mu(n)=0 for 48,49,50; 98,99,100; 124,125,126; ...

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

Union of A070258, A063838 and A070268. Cf. A008683, A070284.

Cf. A261890.

a114180 n = a114180_list !! (n-1)
a114180_list = filter ((== 0) . a261890) [1..]
-- Reinhard Zumkeller, Sep 05 2015

SequencePosition[MoebiusMu[Range[1200]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Jul 23 2023 *)

cp's OEIS Frontend

A055615 a(n) = n * mu(n), where mu is the Möbius function A008683.

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A261869 First differences of A055615.

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A114180 Numbers n with mu(n) = mu(n+1) = mu(n+2).

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Programs

Haskell

Mathematica