A061020 -id:A061020 - cp's OEIS frontend

A344440 a(n) = n + A061020(n).

2, 1, 1, 7, 1, 8, 1, 3, 16, 14, 1, 6, 1, 20, 23, 27, 1, 11, 1, 8, 33, 32, 1, 34, 46, 38, 7, 10, 1, 22, 1, 11, 53, 50, 59, 57, 1, 56, 63, 60, 1, 30, 1, 14, 17, 68, 1, 26, 92, 29, 83, 16, 1, 74, 95, 86, 93, 86, 1, 84, 1, 92, 21, 107, 113, 46, 1, 20, 113, 46, 1, 37, 1, 110, 33, 22, 137, 54, 1, 36, 142, 122, 1, 120, 149

Offset: 1

Antti Karttunen, May 23 2021

Compare the scatter plot to that of A061020.

Antti Karttunen, Table of n, a(n) for n = 1..12427
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A061020, A344439, A344441.

With[{nn = 85}, MapIndexed[First[#2] + #1 &, Drop[CoefficientList[Series[Sum[LiouvilleLambda[k] k x^k/(1 - x^k), {k, 1, #}], {x, 0, #}], x], 1] &[nn]]] (* Michael De Vlieger, May 24 2021, after Jean-François Alcover at A061020 *)

A061020(n) = {my(f=factorint(n)); prod(k=1, #f[, 2], ((-f[k, 1])^(f[k, 2]+1)-1)/(-f[k, 1]-1))}; \\ From A061020
A344440(n) = (n+A061020(n));

a(n) = n + A061020(n).

a(n) = A344439(n) + A344441(n).

A344441 a(n) = A061020(n) + abs(A061020(n)).

Original entry on oeis.org

2, 0, 0, 6, 0, 4, 0, 0, 14, 8, 0, 0, 0, 12, 16, 22, 0, 0, 0, 0, 24, 20, 0, 20, 42, 24, 0, 0, 0, 0, 0, 0, 40, 32, 48, 42, 0, 36, 48, 40, 0, 0, 0, 0, 0, 44, 0, 0, 86, 0, 64, 0, 0, 40, 80, 60, 72, 56, 0, 48, 0, 60, 0, 86, 96, 0, 0, 0, 88, 0, 0, 0, 0, 72, 0, 0, 120, 0, 0, 0, 122, 80, 0, 72, 128, 84, 112, 100, 0, 56, 144

Offset: 1

Antti Karttunen, May 23 2021

nonn

a(k) = 0 for k in A026424. - David A. Corneth, May 23 2021

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

Cf. A026424, A061020, A206369, A344439, A344441.

With[{nn = 91}, Map[# + Abs[#] &, Drop[CoefficientList[Series[Sum[LiouvilleLambda[k] k x^k/(1 - x^k), {k, 1, #}], {x, 0, #}], x], 1] &[nn]]] (* Michael De Vlieger, May 24 2021, after Jean-François Alcover at A061020 *)

A061020(n) = {my(f=factorint(n)); prod(k=1, #f[, 2], ((-f[k, 1])^(f[k, 2]+1)-1)/(-f[k, 1]-1))};
A344441(n) = { my(u=A061020(n)); (u+abs(u)); };

a(n) = A061020(n) + A206369(n) = A061020(n) + abs(A061020(n)).

a(n) = A344440(n) - A344439(n).

A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 5, 7, 4, 10, 6, 12, 6, 8, 11, 16, 7, 18, 12, 12, 10, 22, 10, 21, 12, 20, 18, 28, 8, 30, 21, 20, 16, 24, 21, 36, 18, 24, 20, 40, 12, 42, 30, 28, 22, 46, 22, 43, 21, 32, 36, 52, 20, 40, 30, 36, 28, 58, 24, 60, 30, 42, 43, 48, 20, 66, 48, 44, 24, 70, 35

Offset: 1

N. J. A. Sloane, Feb 06 2012

For more information see the Comments in A061020.
a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a perfect square. For example, a(12) = 6 because |{1,4,5,7,8,11}|=6 and the respective GCDs with 12 are 1,4,1,1,4,1, which are squares. - Geoffrey Critzer, Feb 16 2015
If m is squarefree (A005117), then a(m) = A000010(m) where A000010 is the Euler totient function. - Michel Marcus, Nov 08 2017
Also it appears that the primorials (A002110) is the sequence of indices of minimum records for a(n)/n, and these records are A038110(n)/A060753(n). - Michel Marcus, Nov 09 2017
Also called rho(n). When rho(n) | n, then n is called k-imperfect, with k = n/rho(n), cf. A127724. - M. F. Hasler, Feb 13 2020

P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 25.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Cf. A061020, A206368.
Cf. A027748 row, A124010, A206475 (first differences).
Cf. A078429.
Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect).

a206369 n = product $
   zipWith h (a027748_row n) (map toInteger $ a124010_row n) where
           h p e = sum $ take (fromInteger e + 1) $
                         iterate ((* p) . negate) (1 - 2 * (e `mod` 2))
-- Reinhard Zumkeller, Feb 08 2012

a:= n-> mul(add(i[1]^(i[2]-j)*(-1)^j, j=0..i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2017

Table[Length[Select[Range[n], IntegerQ[GCD[n, #]^(1/2)] &]], {n, 72}] (* Geoffrey Critzer, Feb 16 2015 *)
a[n_] := n*DivisorSum[n, LiouvilleLambda[#]/#&]; Array[a, 72] (* Jean-François Alcover, Dec 04 2017, after Enrique Pérez Herrero *)
f[p_,e_] := Sum[(-1)^(e-k)*p^k, {k,0,e}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

a(n) = sum(k=1, n, issquare(gcd(n, k)));

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
a(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Dec 27 2016

a(n) = sumdiv(n, d, eulerphi(n/d) * issquare(d)); \\ Daniel Suteu, Jun 27 2018

apply( {A206369(n)=vecprod([f[1]^(f[2]+1)\/(f[1]+1)|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Feb 13 2020

from math import prod
from sympy import factorint
def A206369(n): return prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1),p+1)) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 05 2024

a(n) = abs(A061020(n)).
a(n) = n*Sum_{d|n} lambda(d)/d, where lambda(n) is A008836(n). - Enrique Pérez Herrero, Sep 23 2012
Dirichlet g.f.: zeta(s - 1)*zeta(2*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
From Michel Marcus, Nov 05 2017: (Start)
a(2^n) = A001045(n+1);
a(3^n) = A015518(n+1);
a(5^n) = A015531(n+1);
a(7^n) = A015552(n+1);
a(11^n) = A015592(n+1). (End)
a(p^k) = p^k - a(p^(k - 1)) for k > 0 and prime p. - David A. Corneth, Nov 09 2017
a(n) = Sum_{d|n, d is a perfect square} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018
a(p^k) = A071324(p^k), for k >= 0 and prime p. - Michel Marcus, Aug 11 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 30. - Vaclav Kotesovec, Feb 07 2019
G.f.: Sum_{k>=1} lambda(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 23 2019
a(n) = Sum_{i=1..n} A010052(gcd(n,i)). - Ridouane Oudra, Nov 24 2019
a(p^k) = round(p^(k+1)/(p+1)). - M. F. Hasler, Feb 13 2020

A076752 a(n) = Sum_{d is a square divisor of n} n/d.

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 10, 10, 10, 11, 15, 13, 14, 15, 21, 17, 20, 19, 25, 21, 22, 23, 30, 26, 26, 30, 35, 29, 30, 31, 42, 33, 34, 35, 50, 37, 38, 39, 50, 41, 42, 43, 55, 50, 46, 47, 63, 50, 52, 51, 65, 53, 60, 55, 70, 57, 58, 59, 75, 61, 62, 70, 85, 65, 66, 67, 85, 69, 70, 71

Offset: 1

Vladeta Jovovic, Nov 12 2002

The Mobius transform of this sequence appears to generate the sequence of absolute terms of A061020. - R. J. Mathar, Feb 08 2011

			a(8) = 10 as the square divisors of 8 are 1 and 4, and 8/1 + 8/4 = 10. - _David A. Corneth_, Nov 03 2017

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A008836, A035316.

Table[Total[n/Select[Divisors[n], IntegerQ@Sqrt@# &]], {n, 71}] (* Ivan Neretin, Sep 20 2017 *)
Table[DivisorSum[n, n/# &, IntegerQ@ Sqrt@ # &], {n, 71}] (* Michael De Vlieger, Nov 03 2017 *)
f[p_, e_] := p^(k = If[EvenQ[e], 0, 1])*(p^(e + 2 - k) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

a(n,f=factor(n))=prod(i=1,#f~, if(f[i,2]%2, f[i,1]*(f[i,1]^(f[i,2]+1)-1), (f[i,1]^(f[i,2]+2)-1))/(f[i,1]^2-1)) \\ Charles R Greathouse IV, Sep 20 2017

a(n) = sumdiv(n, d, (n/d)*issquare(d)); \\ Michel Marcus, Nov 02 2017

Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) for even e and a(p^e) = p*(p^(e+1)-1)/(p^2-1) for odd e.
a(p ^ (m + 1)) = p * a(p^m) for even m and a(p ^ (m + 1)) = p * a(p^m) + 1 for odd m. - David A. Corneth, Nov 03 2017
a(n) = (lambda * sigma)(n) = (A008836 * A000203)(n), where * is the Dirichlet convolution. - Yuyang Zhao, Nov 02 2017
From Vaclav Kotesovec, Feb 04 2019: (Start)
Dirichlet g.f.: zeta(2*s)*zeta(s-1).
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 180. (End)
G.f.: Sum_{k>=1} x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 19 2021

A127724 k-imperfect numbers for some k >= 1.

Original entry on oeis.org

1, 2, 6, 12, 40, 120, 126, 252, 880, 2520, 2640, 10880, 30240, 32640, 37800, 37926, 55440, 75852, 685440, 758520, 831600, 2600640, 5533920, 6917400, 9102240, 10281600, 11377800, 16687440, 152182800, 206317440, 250311600, 475917120, 715816960, 866829600

Offset: 1

T. D. Noe, Jan 25 2007

For prime powers p^e, define a multiplicative function rho(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e. A number n is called k-imperfect if there is an integer k such that n = k*rho(n). Sequence A061020 gives a signed version of the rho function. As with multiperfect numbers (A007691), 2-imperfect numbers are also called imperfect numbers. As shown by Iannucci, when rho(n) is prime, there is sometimes a technique for generating larger imperfect numbers.
Zhou and Zhu find 5 more terms, which are in the b-file. - T. D. Noe, Mar 31 2009
Does this sequence follow Benford's law? - David A. Corneth, Oct 30 2017
If a term t has a prime factor p from A065508 with exponent 1 and does not have the corresponding prime factor q from A074268, then t*p*q is also a term. - Michel Marcus, Nov 22 2017
For n >= 1, the least n-imperfect numbers are 1, 2, 6, 993803899780063855042560. - Michel Marcus, Feb 13 2018
For any m > 0, if n*p^(2m-1) is k-imperfect, q = rho(p^(2m)) is prime and gcd(pq,n) = 1, then n*p^(2m)*q is also k-imperfect. - M. F. Hasler, Feb 13 2020

			126 = 2*3^2*7, rho(126) = (2-1)*(9-3+1)*(7-1) = 42.  3*42 = 126, so 126 is 3-imperfect. - _Jud McCranie_ Sep 07 2019

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer, 1994, B1.

Giovanni Resta, Table of n, a(n) for n = 1..50 (terms < 10^13, a(1)-a(39) from T. D. Noe (from Iannucci, Zhou, and Zhu), a(40)-a(44) from Donovan Johnson)
David A. Corneth, Conjectured to be the terms up to 10^28
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
Donovan Johnson, 43 terms > 2*10^11
Andrew Lelechenko, 4-imperfect numbers, Apr 19 2014.
Michel Marcus, More 4-imperfect numbers, Nov 07 2017.
Michel Marcus, Least known integers with small denominator-fractional k's, Feb 13 2018.
Allan Wechsler, Some progress in k-imperfect numbers (A127724), Seqfan, Feb 13 2020. Gives a first instance of a 5-imperfect number.
Weiyi Zhou and Long Zhu, On k-imperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.

Cf. A127725 (2-imperfect numbers), A127726 (3-imperfect numbers), A127727 (related primes), A309806 (the k values).
Cf. A061020 (signed version of rho function), A206369 (the rho function).
Cf. A065508, A074268.

f[p_,e_]:=Sum[(-1)^(e-k) p^k, {k,0,e}]; rho[n_]:=Times@@(f@@@FactorInteger[n]); Select[Range[10^6], Mod[ #,rho[ # ]]==0&]

isok(n) = denominator(n/sumdiv(n, d, d*(-1)^bigomega(n/d))) == 1; \\ Michel Marcus, Oct 28 2017

upto(ulim) = {res = List([1]); rhomap = Map(); forprime(p = 2, 3, for(i = 1, logint(ulim, p), mapput(rhomap, p^i, rho(p^i)); iterate(p^i, mapget(rhomap, p^i), ulim))); listsort(res, 1); res}
iterate(m, rhoo, ulim) = {my(c); if(m / rhoo == m \ rhoo, listput(res, m); my(frho = factor(rhoo)); for(i = 1, #frho~, if(m%frho[i, 1] != 0, for(e = 1, logint(ulim \ m, frho[i, 1]), if(mapisdefined(rhomap, frho[i, 1]^e) == 0, mapput(rhomap, frho[i, 1]^e, rho(frho[i, 1]^e))); iterate(m * frho[i, 1]^e, rhoo * mapget(rhomap, frho[i, 1]^e), ulim)); next(2))))}
rho(n) = {my(f = factor(n), res = q = 1); for(i=1, #f~, q = 1; for(j = 1, f[i, 2], q = -q + f[i, 1]^j); res * =q); res} \\ David A. Corneth, Nov 02 2017

A127724_vec=concat(1, select( {is_A127724(n)=!(n%A206369(n))}, [1..10^5]*2))
  /* It is known that the least odd term > 1 is > 10^49. This code defines an efficient function is_A127724, but A127724_vec is better computed with upto(.) */
  A127724(n)=A127724_vec[n] \\ Used in other sequences. - M. F. Hasler, Feb 13 2020

Small correction in name from Michel Marcus, Feb 13 2018

A061019 Negate primes in factorization of n.

Original entry on oeis.org

1, -2, -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, 39, 40, -41, -42, -43, -44, -45, 46, -47, -48, 49, -50, 51, -52, -53, 54, 55, 56, 57, 58, -59, 60, -61, 62, -63, 64, 65, -66, -67, -68, 69, -70

Offset: 1

Marc LeBrun, Apr 13 2001

Inverse Moebius transform of A158523. - Corrected by Antti Karttunen, Nov 26 2024

			a(6) = (-2)(-3) = +6, while a(8) = (-2)^3 = -8.

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

Cf. A000027, A001222, A061020, A001615, A158523 (Möbius transform).
Cf. A027746.
Cf. A239122 (partial sums).

a061019 1 = 1
a061019 n = product $ map negate $ a027746_row n
-- Reinhard Zumkeller, Feb 08 2012

Table[n (-1)^PrimeOmega[n],{n,70}] (* Harvey P. Dale, Oct 05 2011 *)

a(n) = if( bitand(bigomega(n),1), - n, n ); /* Joerg Arndt, Sep 19 2012 */

from functools import reduce
from operator import ixor
from sympy import factorint
def A061019(n): return -n if reduce(ixor, factorint(n).values(),0)&1 else n # Chai Wah Wu, Dec 20 2022

a(n) = n*lambda(n), where lambda is Liouville's function: A008836.
a(n) = (-1)^(number of primes dividing n)*n = n * (-1)^A001222(n) = n*A008836(n).
Totally multiplicative with a(p) = -p for prime p. [Jaroslav Krizek, Nov 01 2009]
Dirichlet g.f.: zeta(2*s-2)/zeta(s-1). Dirichlet inverse of A055615, all terms turned positive there. - R. J. Mathar, Apr 16 2011
a(n) = Sum_{d|n} lambda(d)*psi(d) = sum_{d|n} A008836(d)* A001615(d) = n/lambda(n). - Enrique Pérez Herrero, Sep 18 2012

A078429 Number of integers k among 1..n for which gcd(k,n) is a cube.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 9, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 18, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 18, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 37, 48, 20, 66, 32, 44, 24, 70, 30, 72, 36, 40, 36

Offset: 1

Vladeta Jovovic, Dec 29 2002

Daniel Suteu, Table of n, a(n) for n = 1..10000
Eckford Cohen, A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where a(n) = Psi_3(n).

Cf. A061020, A206369, A327626 (inv. Mob. Trans.).

Cf. A000010, A000027, A010057, A077834, A077947, A210826.

nn = 76; f[list_, i_] := list[[i]]; a = Table[If[IntegerQ[n^(1/3)], 1, 0], {n, 1, nn}]; b =Table[EulerPhi[n], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 25 2015 *)

a(n) = sum(k=1, n, ispower(gcd(n, k), 3)); \\ Michel Marcus, Feb 25 2015

a(n) = sumdiv(n, d, eulerphi(n/d) * ispower(d, 3)); \\ Daniel Suteu, Jun 27 2018

a(n) is multiplicative.
G.f. for a(p^n), p a prime, is given by 1/(1+x+x^2)/(1-p*x).
a(2^n) = A077947(n), a(3^n) = A077834(n).
a(p) = p-1, a(p^2) = p*(p-1), a(p^3) = p^3-p^2+1, a(p^4) = (p-1)*(p+1)*(p^2-p+1), ...
Dirichlet g.f.: zeta(s - 1)*zeta(3*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
a(n) = Sum_{d|n, d is a perfect cube} phi(n/d), where phi(k) is the Euler totient function. Dirichlet convolution of A000010 and A010057. - Daniel Suteu, Jun 27 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 315. - Vaclav Kotesovec, Feb 07 2019
Dirichlet convolution of A000027 and A210826. - R. J. Mathar, Jun 05 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} A010057(gcd(n,k)).
a(n) = Sum_{k=1..n} A010057(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A118206 Euler transform of the Liouville function.

Original entry on oeis.org

1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 2, 0, -2, -2, 1, 2, 2, -2, -2, 0, 2, -1, -1, -2, 2, 5, 4, -5, -5, -2, 4, 2, -2, -7, 3, 8, 5, -7, -6, 1, 14, 4, -9, -14, 2, 5, 5, -10, -7, 6, 22, 3, -12, -20, 1, 15, 15, -16, -12, 4, 25, 6, -14, -31, 13, 33, 14, -39, -32, -6, 39, 15, -20, -31, 33, 41, 14, -53, -44, 3, 66, 12, -35, -51, 22, 48, 36, -60, -43, 21

Offset: 0

Stuart Clary, Apr 15 2006

Paul D. Hanna, Table of n, a(n) for n = 0..1000
Peter Bala, Cyclotomic polynomials and the generating functions of A118205 and A118206
Wikipedia, Cyclotomic polynomial
Wikipedia, Liouville function

Cf. A118205, A118207, A118208, A061020, A117209.

nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^(-lambda[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ]
max = 100; s = Product[(1 - x^k)^(-LiouvilleLambda[k]), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*moebius(core(d)))*x^m/m)+x*O(x^n)),n)} /* Cf. A061020 - Paul D. Hanna, Sep 22 2011 */

G.f.: A(x) = Product_{k>=1} (1 - x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.
Logarithmic derivative yields A061020. - Paul D. Hanna, Sep 22 2011
G.f.: A(x) = Product_{k >= 1} C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

A158523 Moebius transform of negated primes in factorization of n.

Original entry on oeis.org

1, -3, -4, 6, -6, 12, -8, -12, 12, 18, -12, -24, -14, 24, 24, 24, -18, -36, -20, -36, 32, 36, -24, 48, 30, 42, -36, -48, -30, -72, -32, -48, 48, 54, 48, 72, -38, 60, 56, 72, -42, -96, -44, -72, -72, 72, -48, -96, 56, -90, 72, -84, -54, 108, 72, 96, 80, 90, -60, 144, -62, 96, -96, 96, 84, -144, -68, -108, 96, -144

Offset: 1

Jaroslav Krizek, Mar 20 2009

			a(72) = a(2^3*3^2) = (-1)^3*(2+1)*2^(3-1) * (-1)^2*(3+1)*3^(2-1) = (-12)*12 = -144.

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

Cf. A061019, A008683, A061020, A007427, A000012, A007428, A000005, A001615, A001222, A000040, A006881, A120944, A000961, A378434.

f[p_, e_] := (-1)^e*(p + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 05 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i,2]*(f[i,1]+1)*f[i,1]^(f[i,2]-1));} \\ Amiram Eldar, Jan 05 2023

Multiplicative with a(p^e) = (-1)^e*(p+1)*p^(e-1), e>0. a(1)=1.
a(n) = mu(n) * A061019(n) = A008683(n) * A061019(n) = A061020(n) * A007427(n) = A061020(n) * A007428(n) * A000012(n) = A007427(n) * A000012(n) * A061019(n) = A007428(n) * A000005(n) * A061019(n), where operation * denotes Dirichlet convolution. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d).
Inverse Moebius transform gives A061019.
a(n) = (-1)^A001222(n)*A001615(n).
Apparently the Dirichlet inverse of A048250. - R. J. Mathar, Jul 15 2010
Dirichlet g.f.: zeta(2*s-2)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Jan 05 2023

More terms from Antti Karttunen, Nov 26 2024

A118205 Euler transform of the negative of the Liouville function.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 2, 0, -2, 3, -2, 1, 2, -3, 3, -2, 0, 3, -2, 3, -2, 0, 2, -2, 3, -1, 0, 1, -2, 5, 0, 0, 1, -2, 1, 1, 2, 0, 1, -2, 1, 4, -1, 4, -2, -3, 6, -2, 5, 6, -8, 6, -4, 2, 9, -8, 7, -4, -1, 11, -1, 5, 1, -8, 5, 2, 4, 7, -8, 4, 2, 1, 14, -2, 0, -1, -6, 19, 2, 5, 6, -15, 12, 1, 3, 18, -17, 1, 9, 0, 29, -4, -3, 4, -13, 14, 17, 2, 0, -4

Offset: 0

Stuart Clary, Apr 15 2006

Peter Bala, Cyclotomic polynomials and the generating functions of A118205 and A118206
Wikipedia, Cyclotomic polynomial
Wikipedia, Liouville function

Cf. A061020, A117208, A118206, A118207, A118208.

nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^lambda[k], {k, 1, nmax} ], {x, 0, nmax} ], x ]
(* Second program (needs Mma >= 7.0): *)
nmax = 100;
Product[(1 - x^n)^LiouvilleLambda[n], {n, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& (* Jean-François Alcover, Jan 08 2020 *)

G.f.: A(x) = Product_{k>=1} (1 - x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

G.f.: A(x) = - Product_{k >= 1} C(k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

cp's OEIS Frontend

A344440 a(n) = n + A061020(n).

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A344441 a(n) = A061020(n) + abs(A061020(n)).

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A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

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A076752 a(n) = Sum_{d is a square divisor of n} n/d.

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A127724 k-imperfect numbers for some k >= 1.

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A061019 Negate primes in factorization of n.

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