A061019 -id:A061019 - cp's OEIS frontend

A347236 a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes.

1, 1, 2, 7, 2, 2, 4, 13, 19, 2, 2, 14, 4, 4, 4, 55, 2, 19, 4, 14, 8, 2, 6, 26, 39, 4, 68, 28, 2, 4, 6, 133, 4, 2, 8, 133, 4, 4, 8, 26, 2, 8, 4, 14, 38, 6, 6, 110, 93, 39, 4, 28, 6, 68, 4, 52, 8, 2, 2, 28, 6, 6, 76, 463, 8, 4, 4, 14, 12, 8, 2, 247, 6, 4, 78, 28, 8, 8, 4, 110, 421, 2, 6, 56, 4, 4, 4, 26, 8, 38, 16

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of A003961 and A061019.
Dirichlet convolution of A003973 and A158523.
Multiplicative because A003961 and A061019 are.
All terms are positive because all terms of A347237 are nonnegative and A347237(1) = 1.
Union of sequences A001359 and A108605 (= 2*A001359) seems to give the positions of 2's in this sequence.

Cf. A000040, A001223, A001359, A003961, A003973, A061019, A108605, A158523, A347237 (Möbius transform), A347238 (Dirichlet inverse), A347239.
Cf. also A347136.
Cf. A151800.

f[p_, e_] := ((np = NextPrime[p])^(e + 1) - (-p)^(e + 1))/(np + p); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 02 2021 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));

a(n) = Sum_{d|n} A003961(n/d) * A061019(d).
a(n) = Sum_{d|n} A003973(n/d) * A158523(d).
a(n) = Sum_{d|n} A347237(d).
a(n) = A347239(n) - A347238(n).
For all n >= 1, a(A000040(n)) = A001223(n).
Multiplicative with a(p^e) = (A151800(p)^(e+1)-(-p)^(e+1))/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A239122 Partial sums of A061019.

Original entry on oeis.org

1, -1, -4, 0, -5, 1, -6, -14, -5, 5, -6, -18, -31, -17, -2, 14, -3, -21, -40, -60, -39, -17, -40, -16, 9, 35, 8, -20, -49, -79, -110, -142, -109, -75, -40, -4, -41, -3, 36, 76, 35, -7, -50, -94, -139, -93, -140, -188, -139, -189, -138, -190, -243, -189, -134

Offset: 1

Reinhard Zumkeller, Mar 10 2014

sign

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

Cf. A002819.

a239122 n = a239122_list !! (n-1)
a239122_list = scanl1 (+) a061019_list

Accumulate[Table[n*(-1)^PrimeOmega[n],{n,70}]] (* Harvey P. Dale, Apr 02 2015 *)

from functools import reduce
from operator import ixor
from sympy import factorint
def A239122(n): return sum(-i if reduce(ixor, factorint(i).values(), 0)&1 else i for i in range(1,n+1)) # Chai Wah Wu, Jan 03 2023

A061020 Negate primes in factorizations of divisors of n, then sum.

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

Marc LeBrun, Apr 13 2001

Analog of sigma function A000203(n) with primes negated.

Unsigned sequence |a(n)| (A206369) gives the number of numbers 1 <= k <= n for which GCD(k,n) is a square. |a(n)| = Sum_{d|n} d*(-1)^bigomega(n/d). - Vladeta Jovovic, Dec 29 2002

			a(12) = 1-2-3+4+6-12 = (1-2+4)*(1-3) = -6.

T. D. Noe, Table of n, a(n) for n = 1..10000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, Involve, a Journal of Mathematics, Vol. 15, No. 2 (2022), pp. 251-270; arXiv preprint, arXiv:2012.04625 [math.CO], 2020-2021.
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Cf. A000203, A061019, A076792, A206369.

Cf. A027750, A007913.

a061020 = sum . map a061019 . a027750_row
-- Reinhard Zumkeller, Feb 08 2012

with(numtheory):
A061020 := proc(n) local d; add(d*(-1)^bigomega(d), d=divisors(n)) end:
seq(A061020(n), n=1..72); # Peter Luschny, Aug 29 2013

nmax = 72; Drop[ CoefficientList[ Series[ Sum[ LiouvilleLambda[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* Stuart Clary, Apr 15 2006, updated by Jean-François Alcover, Dec 04 2017 *)
f[p_, e_] := ((-p)^(e+1)-1)/(-p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

for(n=1,100,print1(sumdiv(n,d,(d)*moebius(core(d))),","))

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1+p*X))[n]) \\ Ralf Stephan

A061020(n) = {my(f=factorint(n)); prod(k=1, #f[,2], ((-f[k,1])^(f[k,2]+1)-1)/(-f[k,1]-1))} \\ Andrew Lelechenko, Apr 22 2014

Replace each divisor d of n by A061019[d] and sum. Replace p^q with (1-(-p)^(q+1))/(1+p) in prime factorization of n.
Inverse mobius transform of A061019. In other words a(n) = Sum_{d|n} d*(-1)^bigomega(d), where bigomega(n) = A001222(n).
a(n) = Sum_{d|n} d*mu(core(d)) where core(x) = A007913(x) is the smallest number such that x*core(x) is a square. - Benoit Cloitre, Apr 07 2002
G.f.: A(x) = Sum_{k>=1} lambda(k)*k*x^k/(1 - x^k) where lambda(k) is the Liouville function, A008836. - Stuart Clary, Apr 15 2006
G.f.: A(x) is x times the logarithmic derivative of A118206(x). - Stuart Clary, Apr 15 2006
Dirichlet g.f.: zeta(s)*zeta(2 s - 2)/zeta(s - 1). - Stuart Clary, Apr 15 2006
a(n) = Sum_{d|n} d*lambda(d), where lambda(n) is A008836(n). - Enrique Pérez Herrero, Aug 29 2013

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

A347238 Dirichlet inverse of A347236.

Original entry on oeis.org

1, -1, -2, -6, -2, 2, -4, 0, -15, 2, -2, 12, -4, 4, 4, 0, -2, 15, -4, 12, 8, 2, -6, 0, -35, 4, 0, 24, -2, -4, -6, 0, 4, 2, 8, 90, -4, 4, 8, 0, -2, -8, -4, 12, 30, 6, -6, 0, -77, 35, 4, 24, -6, 0, 4, 0, 8, 2, -2, -24, -6, 6, 60, 0, 8, -4, -4, 12, 12, -8, -2, 0, -6, 4, 70, 24, 8, -8, -4, 0, 0, 2, -6, -48, 4, 4, 4, 0, -8

Offset: 1

Antti Karttunen, Aug 24 2021

Multiplicative because A347236 is.
It seems that A046099 gives the positions of zeros.
This follows from the formula for a(p^e). - Sebastian Karlsson, Sep 01 2021

Cf. A000040, A001223, A003961, A046099, A061019, A347236, A347239.

Cf. A151800.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));
v347238 = DirInverseCorrect(vector(up_to,n,A347236(n)));
A347238(n) = v347238[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A347236(n/d) * a(d).
a(n) = A347239(n) - A347236(n).
For all n >= 1, a(A000040(n)) = -A001223(n).
Multiplicative with a(p^e) = p - A151800(p) if e = 1, -p*A151800(p) if e = 2 and 0 if e > 2. - Sebastian Karlsson, Sep 01 2021

A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, -3, -2, 5, -4, 6, -6, -11, 7, 12, -10, -10, -12, 18, 8, 21, -16, -21, -18, -20, 12, 30, -22, 22, 21, 36, -20, -30, -28, -24, -30, -43, 20, 48, 24, 35, -36, 54, 24, 44, -40, -36, -42, -50, -28, 66, -46, -42, 43, -63, 32, -60, -52, 60, 40, 66, 36, 84, -58, 40, -60, 90, -42, 85, 48, -60, -66, -80, 44, -72, -70, -77, -72, 108, -42

Offset: 1

Stuart Clary, Apr 15 2006

Related to the logarithmic derivative of A118207(x) and A118208(x).

Related to a signed variant of A022998 via Mobius inversion. - R. J. Mathar, Jul 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008836, A022998, A028260, A061019, A061020, A062157, A117212, A118207, A118208.

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; Drop[ CoefficientList[ Series[ Sum[ lambda[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ]
f[p_, e_] := (p*(-p)^e+1)/(p+1); f[2, e_] := ((-1)^e*2^(e+2) - 1)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 12 2023 *)

a(n) = sumdiv(n, d, (-1)^(n/d - 1)*(-1)^vecsum(factor(d)[,2])*d) \\ Michel Marcus, Dec 10 2016

a(n) = Sum_{d|n} (-1)^(n/d - 1)*lambda(d)*d, Dirichlet convolution of A061019 and A062157.
G.f.: A(x) is x times the logarithmic derivative of A118207(x).
G.f.: A(x) = A061020(x) - 2 A061020(x^2).
Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-2^(1-s))/zeta(s-1). - R. J. Mathar, Jul 03 2011
a(n) > 0 for n in A028260. - Michel Marcus, Dec 10 2016
Multiplicative with a(2^e) = ((-1)^e*2^(e+2) - 1)/3, and a(p^e) = (p*(-p)^e+1)/(p+1) for an odd prime p. - Amiram Eldar, Aug 12 2023

A347237 Möbius transform of A347236.

Original entry on oeis.org

1, 0, 1, 6, 1, 0, 3, 6, 17, 0, 1, 6, 3, 0, 1, 42, 1, 0, 3, 6, 3, 0, 5, 6, 37, 0, 49, 18, 1, 0, 5, 78, 1, 0, 3, 102, 3, 0, 3, 6, 1, 0, 3, 6, 17, 0, 5, 42, 89, 0, 1, 18, 5, 0, 1, 18, 3, 0, 1, 6, 5, 0, 51, 330, 3, 0, 3, 6, 5, 0, 1, 102, 5, 0, 37, 18, 3, 0, 3, 42, 353, 0, 5, 18, 1, 0, 1, 6, 7, 0, 9, 30, 5, 0, 3, 78, 3, 0, 17

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of A003972 (prime shifted phi) with A061019.
Dirichlet convolution of A003961 with A158523.
Multiplicative because A003972 and A061019 are, and also because A347236 is.
From Antti Karttunen, Aug 25 2021: (Start)
All terms are nonnegative because sequence is multiplicative and a(p^k) >= 0 for all primes p and k >= 0.
Proof: For any prime p, sequence a(p^k), k>=0, is obtained as an ordinary convolution of sequences (-p)^k and the first differences of q^k, where q = A151800(p). (E.g., for powers of 2, the sequences convolved are A122803 and A025192, giving A102901.) This convolution is an alternating sum, with the terms 1*(q-1)*q^(k-1), -(p)*(q-1)*q^(k-2), (p^2)*(q-1)*q^(k-3), -(p^3)*(q-1)*q^(k-4), ..., (p^(k-1))*(q-1), -(p^k), for odd k, with sum of each consecutive pair being nonnegative because q >= p+1, while with an even exponent k, the leftover term p^k at the end is also positive, thus the whole sum is nonnegative also in that case.
(End)

Cf. A000040, A001223, A003961, A003972, A008683, A008836, A016825 (positions of zeros), A061019, A102901, A120612, A158523, A347236.
Cf. also A347137.
Cf. A151800.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));
\\ Or alternatively as:
A158523(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((-1)^e)*(p+1)*(p^(e-1))); };
A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

a(n) = Sum_{d|n} A008683(n/d) * A347236(d).
a(n) = Sum_{d|n} A003972(n/d) * A061019(d).
a(n) = Sum_{d|n} A003961(n/d) * A158523(d).
For all n >= 1, a(A000040(n)) = A001223(n) - 1.
For all n >= 0, a(2^n) = A102901(n).
For all n >= 0, a(3^n) = A120612(n).
Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A347239 Sum of A347236 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 13, 4, 4, 0, 26, 0, 8, 8, 55, 0, 34, 0, 26, 16, 4, 0, 26, 4, 8, 68, 52, 0, 0, 0, 133, 8, 4, 16, 223, 0, 8, 16, 26, 0, 0, 0, 26, 68, 12, 0, 110, 16, 74, 8, 52, 0, 68, 8, 52, 16, 4, 0, 4, 0, 12, 136, 463, 16, 0, 0, 26, 24, 0, 0, 247, 0, 8, 148, 52, 16, 0, 0, 110, 421, 4, 0, 8, 8, 8, 8, 26, 0, 8, 32

Offset: 1

Antti Karttunen, Aug 24 2021

nonn

It seems that A030059 gives the positions of all zeros.

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

Cf. A000290, A001223, A001248, A003961, A061019, A030059, A030229, A347236, A347238.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));
v347238 = DirInverseCorrect(vector(up_to,n,A347236(n)));
A347238(n) = v347238[n];
A347239(n) = (A347236(n)+A347238(n));

a(n) = A347236(n) + A347238(n).
a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A347236(d) * A347238(n/d).
For all n >= 1, a(A030059(n)) = 0 and a(A030229(n)) = 2*A347236(A030229(n)).
For all n >= 1, a(A001248(n)) = A000290(A001223(n)).

A358272 Multiplicative sequence with a(p^e) = (-1)^e * p^(2*floor(e/2)) for prime p and e >= 0.

Original entry on oeis.org

1, -1, -1, 4, -1, 1, -1, -4, 9, 1, -1, -4, -1, 1, 1, 16, -1, -9, -1, -4, 1, 1, -1, 4, 25, 1, -9, -4, -1, -1, -1, -16, 1, 1, 1, 36, -1, 1, 1, 4, -1, -1, -1, -4, -9, 1, -1, -16, 49, -25, 1, -4, -1, 9, 1, 4, 1, 1, -1, 4, -1, 1, -9, 64, 1, -1, -1, -4, 1, -1, -1, -36, -1, 1, -25, -4, 1, -1, -1, -16

Offset: 1

Werner Schulte, Nov 07 2022

Signed version of A008833.

Cf. A000010, A008833, A008836, A034444, A061019.

A358272 := proc(n)
    local a,pe,e,p ;
    a := 1;
    for pe in ifactors(n)[2] do
        e := op(2,pe) ;
        p := op(1,pe) ;
        a := a*(-1)^e*p^(2*floor(e/2)) ;
    end do:
    a ;
end proc:
seq(A358272(n),n=1..80) ; # R. J. Mathar, Jan 17 2023

f[p_, e_] := (-1)^e * p^(2*Floor[e/2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 07 2022 *)

from math import prod
from sympy import factorint
def A358272(n): return prod(-p**(e&-2) if e&1 else p**(e&-2) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 17 2023

a(n) = lambda(n) * A008833(n) for n > 0 where lambda(n) = A008836(n).
Dirichlet g.f.: zeta(2*s-2) / zeta(s).
Dirichlet inverse b(n), n > 0, is multiplicative with b(p) = 1 and b(p^e) = 1 - p^2 for prime p and e > 1.
Dirichlet convolution with A034444 equals A008833.
Equals Dirichlet convolution of A000010 and A061019.
Conjecture: a(n) = Sum_{k=1..n} gcd(k, n) * lambda(gcd(k, n)) for n > 0.
a(n) = Sum_{d|n} lambda(d)*d*phi(n/d), where lambda(n) = A008836(n). - Ridouane Oudra, May 11 2025

cp's OEIS Frontend

A347236 a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes.

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A239122 Partial sums of A061019.

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A061020 Negate primes in factorizations of divisors of n, then sum.

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A158523 Moebius transform of negated primes in factorization of n.

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A347238 Dirichlet inverse of A347236.

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A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

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A347237 Möbius transform of A347236.

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