A101035 -id:A101035 - cp's OEIS frontend

A127140 Square of triangle A127139, row sums = A101035.

1, -4, 1, -6, 0, 1, 4, -4, 0, 1, -10, 0, 0, 0, 1, 24, -6, -4, 0, 0, 1, -14, 0, 0, 0, 0, 0, 1, 0, 4, 0, -4, 0, 0, 0, 1, 9, 0, -6, 0, 0, 0, 0, 0, 1, 40, -10, 0, 0, -4, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Row sums = A101035: (1, -3, -5, 1, -9, 15, ...). A127140 * [1, 2, 3, ...] = A055615: (1, -2, -3, 0, -5, 6, -7, ...).

			First few rows of the triangle:
    1;
   -4,  1;
   -6,  0,  1;
    4, -4,  0,  1;
  -10,  0,  0,  0, 1;
   24, -6, -4,  0, 0, 1;
  -14,  0,  0,  0, 0, 0, 1;
    0,  4,  0, -4, 0, 0, 0, 1;
    9,  0, -6,  0, 0, 0, 0, 0, 1;
  ...

Cf. A127139, A101035, A055615.

Square of A127139.

A129667 Dirichlet inverse of the Abelian group count (A000688).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, 1, 1, -1, 0, -1, 1, 0, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, 0, -1, -1, -1, 1, 1, 1, -1, 0, -1, 1, 1, 1, -1, 0, 1, 0, 1, 1, -1, -1, -1, 1, 1, 0, 1, -1, -1, 1, 1, -1, -1, 0, -1, 1, 1, 1, 1, -1, -1, 0, 0, 1, -1, -1, 1, 1, 1, 0, -1, -1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 0, -1, 1, -1

Offset: 1

Gerard P. Michon, Apr 28 2007, May 01 2007

The simple formula which gives the value of this multiplicative function for the power of any prime can be derived from Euler's celebrated "Pentagonal Number Theorem" (applied to the generating function of the partition function A000041 on which A000688 is based).

			a(8) and a(27) are zero because the sequence vanishes for the cubes of primes. Not so with fifth powers of primes (since 5 is a pentagonal number) so a(32) is nonzero.

R. J. Mathar, Table of n, a(n) for n = 1..1000
Gérard P. Michon, Multiplicative Functions.
Gérard P. Michon, Partition Function and Pentagonal Numbers.

Cf. A000041, A000326, A000688, A005449, A023900, A101035.

A000326inv := proc(n)
    local x,a ;
    for x from 0 do
        a := x*(3*x-1)/2 ;
        if a > n then
            return -1 ;
        elif a = n then
            return x;
        end if;
    end do:
end proc:
A005449inv := proc(n)
    local x,a ;
    for x from 0 do
        a := x*(3*x+1)/2 ;
        if a > n then
            return -1 ;
        elif a = n then
            return x;
        end if;
    end do:
end proc:
A129667 := proc(n)
    local a,e1,e2 ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        e1 := A000326inv(op(2,pe)) ;
        e2 := A005449inv(op(2,pe)) ;
        if e1 >= 0 then
            a := a*(-1)^e1 ;
        elif e2 >= 0 then
            a := a*(-1)^e2 ;
        else
            a := 0 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Nov 24 2017

a[n_] := a[n] = If[n == 1, 1, -Sum[FiniteAbelianGroupCount[n/d] a[d], {d, Most @ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 16 2020 *)

Multiplicative function for which a(p^e) either vanishes or is equal to (-1)^m, for any prime p, if e is either m(3m-1)/2 or m(3m+1)/2 (these integers are the pentagonal numbers of the first and second kind, A000326 and A005449).
Dirichlet g.f.: 1 / Product_{k>=1} zeta(k*s). - Ilya Gutkovskiy, Nov 06 2020
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Product_{p prime} ((1-1/p) * (1 + Sum_{m>=1} (1/p^(m*(3*m-1)/2) + 1/p^(m*(3*m+1)/2)))) = 0.85358290653064143678... . - Amiram Eldar, Feb 17 2024

A328722 Dirichlet g.f.: 1 / zeta(s-1)^2.

Original entry on oeis.org

1, -4, -6, 4, -10, 24, -14, 0, 9, 40, -22, -24, -26, 56, 60, 0, -34, -36, -38, -40, 84, 88, -46, 0, 25, 104, 0, -56, -58, -240, -62, 0, 132, 136, 140, 36, -74, 152, 156, 0, -82, -336, -86, -88, -90, 184, -94, 0, 49, -100, 204, -104, -106, 0, 220, 0, 228, 232, -118, 240

Offset: 1

Ilya Gutkovskiy, Oct 26 2019

Dirichlet inverse of A038040.
Dirichlet convolution of A055615 with itself.
Moebius transform of A101035.

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

Cf. A007427, A008683, A038040, A046099 (positions of 0's), A055615, A101035.

a[1] = 1; a[n_] := -Sum[(n/d) DivisorSigma[0, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 60}]
Table[n DivisorSum[n, MoebiusMu[n/#] MoebiusMu[#] &], {n, 1, 60}]
f[p_, e_] := Switch[e, 1, -2*p, 2, p^2, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

A007427(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d))); \\ From A007427
A328722(n) = (n*A007427(n)); \\ Antti Karttunen, Nov 15 2021

a(1) = 1; a(n) = -Sum_{d|n, dA038040(n/d) * a(d).
a(n) = n * A007427(n).
a(n) = Sum_{d|n} mu(n/d) * A101035(d).
Multiplicative with a(p) = -2*p, a(p^2) = p^2, and a(p) = 0 for e >= 3. - Amiram Eldar, Sep 15 2023

A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 0, -1, 0, -3, 2, -5, 0, -2, 2, -9, 4, -11, 2, 5, 0, -15, 2, -17, 4, 7, 2, -21, 8, -6, 2, -4, 4, -27, -2, -29, 0, 11, 2, 17, 8, -35, 2, 13, 8, -39, -6, -41, 4, 8, 2, -45, 16, -10, -2, 17, 4, -51, 0, 29, 8, 19, 2, -57, 4, -59, 2, 12, 0, 35, -14, -65, 4, 23, -10, -69, 24, -71, 2, 4, 4, 47, -18, -77, 16, -8, 2, -81, -4, 47, 2, 29, 8, -87, 4

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323912, Sequence Machine.
Wikipedia, Dirichlet convolution.

Cf. A083254, A323913.

Sequences that appear in the convolution formulas: A002033, A023900, A046692, A055615, A067824, A074206, A101035, A130054, A174725, A191161, A253249, A323910 (Möbius transform), A328722, A330575.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];

A083254(n) = (2*eulerphi(n)-n);
memoA323912 = Map();
A323912(n) = if(1==n,1,my(v); if(mapisdefined(memoA323912,n,&v), v, v = -sumdiv(n,d,if(dA083254(n/d)*A323912(d),0)); mapput(memoA323912,n,v); (v))); \\ Antti Karttunen, Nov 22 2024

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083254(n/d) * a(d).
From Antti Karttunen, Nov 22 2024: (Start)
Following convolution formulas were conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly. The first one is certainly true, because A083254 is Möbius transform of A033879:
a(n) = Sum_{d|n} A323910(d).
a(n) = Sum_{d|n} A023900(d)*A074206(n/d) = Sum_{d|n} A002033(d-1)*A023900(n/d).
a(n) = Sum_{d|n} A055615(d)*A067824(n/d)
a(n) = Sum_{d|n} A046692(d)*A253249(n/d)
a(n) = Sum_{d|n} A130054(d)*A174725(n/d)
a(n) = Sum_{d|n} A101035(d)*A330575(n/d)
a(n) = Sum_{d|n} A191161(d)*A328722(n/d)
(End)

A327960 Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1)^2).

Original entry on oeis.org

1, -5, -7, 8, -11, 35, -15, -4, 15, 55, -23, -56, -27, 75, 77, 0, -35, -75, -39, -88, 105, 115, -47, 28, 35, 135, -9, -120, -59, -385, -63, 0, 161, 175, 165, 120, -75, 195, 189, 44, -83, -525, -87, -184, -165, 235, -95, 0, 63, -175, 245, -216, -107, 45, 253, 60, 273, 295, -119, 616

Offset: 1

Ilya Gutkovskiy, Oct 22 2019

Dirichlet inverse of A060640.

Moebius transform applied twice to A101035.

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

Cf. A046101 (positions of 0's), A046692, A055615, A060640, A101035.

a[1] = 1; a[n_] := -Sum[Sum[j DivisorSigma[0, j], {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 60}]
f[p_, e_] := Which[e==1, -(2*p+1), e==2, p^2+2*p, e==3, -p^2, e>3, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(1) = 1; a(n) = -Sum_{d|n, dA060640(n/d) * a(d).
a(n) = Sum_{d|n} A046692(n/d) * A055615(d).
Multiplicative with a(p^e) = -(2*p+1) if e=1, p^2+2*p if e=2, -p^2 if e=3, and 0 otherwise. - Amiram Eldar, Dec 02 2020

A120630 Dirichlet inverse of A002654.

Original entry on oeis.org

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

Offset: 1

Gerard P. Michon, Jun 25 2006

			a(65)=4 because 65 is 5 times 13 and both of those primes are congruent to 1 modulo 4. Doubling an odd index yields the opposite of the value (e.g., a(130)=-4) because a(2)=-1. Doubling an even index yields zero.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002654, A006752, A023900, A046692, A053822, A053825, A053826, A101035.

A120630 := proc(n)
    local a,pp;
    if n = 1 then
        1;
    else
        a := 1 ;
        for pp in ifactors(n)[2] do
            if op(2,pp) > 2 then
                a := 0;
            elif op(1,pp) = 2 then
                if op(2,pp) = 1 then
                    a := -a ;
                else
                    a := 0 ;
                end if;
            elif modp(op(1,pp),4) = 3 then
                if op(2,pp) = 1 then
                    a := 0 ;
                else
                    a := -a ;
                end if;
            else
                if op(2,pp) = 1 then
                    a := -2*a ;
                else
                    ;
                end if;
            end if;
        end do:
        a;
    end if;
end proc: # R. J. Mathar, Sep 15 2015

A120630[n_] := Module[{a, pp}, If[n == 1, 1, a = 1; Do[Which[pp[[2]] > 2, a = 0, pp[[1]] == 2, If[pp[[2]] == 1, a = -a, a = 0], Mod[pp[[1]], 4] == 3, If[pp[[2]] == 1, a = 0, a = -a], True, If[pp[[2]] == 1, a = -2*a]], {pp, FactorInteger[n]}]; a]]; Array[A120630, 120] (* Jean-François Alcover, Apr 24 2017, after R. J. Mathar *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv( n, d, (d%4==1) - (d%4==3))))} \\ Andrew Howroyd, Aug 05 2018

Multiplicative function with a(p^e)=0 if e>2. a(2)=-1, a(4)=0. If p is a prime congruent to 3 modulo 4, then a(p)=0 and a(p^2)=-1. If p is a prime congruent to 1 modulo 4, then a(p)=-2 and a(p^2)=1.

Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 3/(2*Pi*G) = 0.521269..., and G is Catalan's constant (A006752). - Amiram Eldar, Jan 22 2024

A328502 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).

Original entry on oeis.org

1, -3, -7, -2, -21, 21, -43, -4, -12, 63, -111, 14, -157, 129, 147, -8, -273, 36, -343, 42, 301, 333, -507, 28, -80, 471, -36, 86, -813, -441, -931, -16, 777, 819, 903, 24, -1333, 1029, 1099, 84, -1641, -903, -1807, 222, 252, 1521, -2163, 56, -252, 240, 1911, 314, -2757, 108, 2331

Offset: 1

Ilya Gutkovskiy, Oct 22 2019

Dirichlet inverse of A057660.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.

Cf. A000010, A008683, A030230 (positions of negative terms), A057660, A101035.

a[1] = 1; a[n_] := -Sum[DivisorSigma[2, (n/d)^2]/DivisorSigma[1, (n/d)^2] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
Table[DivisorSum[n, EulerPhi[n/#] MoebiusMu[#] #^2 &], {n, 1, 55}]
f[p_, e_] := If[e == 1, p - 1 - p^2, -p^(e - 1)*(p - 1)^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

a(n)={sumdiv(n, d, eulerphi(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA057660(n/d) * a(d).
a(n) = Sum_{d|n} phi(n/d) * mu(d) * d^2.
Multiplicative with a(p) = p - 1 - p^2, and a(p^e) = -p^(e-1) * (p-1)^2, for e > 1. - Amiram Eldar, Dec 03 2022
a(n) = Sum_{k = 1..n} gcd(k, n)^2 * mu(gcd(k, n)) (follows from an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

A328641 Dirichlet g.f.: zeta(s)^2 / zeta(s-1)^2.

Original entry on oeis.org

1, -2, -4, -1, -8, 8, -12, 0, 0, 16, -20, 4, -24, 24, 32, 1, -32, 0, -36, 8, 48, 40, -44, 0, 8, 48, 4, 12, -56, -64, -60, 2, 80, 64, 96, 0, -72, 72, 96, 0, -80, -96, -84, 20, 0, 88, -92, -4, 24, -16, 128, 24, -104, -8, 160, 0, 144, 112, -116, -32, -120, 120, 0, 3, 192

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A029935.
Dirichlet convolution of A023900 with itself.
Inverse Moebius transform of A101035.

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

Cf. A023900, A029935, A101035.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[EulerPhi[j], EulerPhi[j], j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 65}]
f[p_, e_] := If[e == 1, 2*(1 - p), (p - 1)*(e*p - p - e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv(n, d, eulerphi(d) * eulerphi(n/d))))} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA029935(n/d) * a(d).
a(n) = Sum_{d|n} A101035(d).
Multiplicative with a(p) = 2*(1-p), and a(p^e) = (p-1)*(e*p-p-e-1) for e > 1. - Amiram Eldar, Dec 03 2022

A347095 Sum of Pillai's arithmetical function (A018804) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 30, 0, 21, 25, 54, 0, 35, 0, 78, 90, 49, 0, 51, 0, 63, 130, 126, 0, 95, 81, 150, 85, 91, 0, 0, 0, 113, 210, 198, 234, 172, 0, 222, 250, 171, 0, 0, 0, 147, 153, 270, 0, 235, 169, 147, 330, 175, 0, 231, 378, 247, 370, 342, 0, 405, 0, 366, 221, 257, 450, 0, 0, 231, 450, 0, 0, 424, 0, 438, 245, 259, 546

Offset: 1

Antti Karttunen, Aug 18 2021

nonn

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A018804, A030059, A030229, A101035.

Cf. also A347094.

up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA018804(n)));
A101035(n) = v101035[n];
A347095(n) = (A018804(n)+A101035(n));

a(n) = A018804(n) + A101035(n).
For n > 1, a(n) = -Sum_{d|n, 1A018804(d) * A101035(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A018804(A030229(n)).

A351402 G.f. A(x) satisfies: 1 / (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).

Original entry on oeis.org

1, 1, -1, -3, -1, 1, 4, 2, -2, -5, 4, 2, -2, -10, 3, 10, 21, -15, -26, -23, 34, 28, 25, -54, -18, 2, 67, -48, -22, -55, 116, 44, 37, -227, -10, 32, 295, -85, -76, -336, 254, 74, 250, -451, 59, -127, 672, -294, -69, -761, 740, 77, 657, -1208, 59, -450, 1700, -487, 241, -1892, 1202

Offset: 0

Ilya Gutkovskiy, Feb 10 2022

sign

Euler transform of A007427.

Cf. A000005, A006171, A007427, A101035, A117209, A351403.

nmax = 60; A007427[n_] := Sum[MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}]; CoefficientList[Series[Product[1/(1 - x^k)^A007427[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f. A(x) satisfies: 1 / (1 - x) = Product_{k>=1} A(x^k)^A000005(k).
G.f.: Product_{k>=1} 1 / (1 - x^k)^A007427(k).
G.f.: exp( Sum_{k>=1} A101035(k) * x^k / k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A101035(k) * a(n-k).

cp's OEIS Frontend

A127140 Square of triangle A127139, row sums = A101035.

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A129667 Dirichlet inverse of the Abelian group count (A000688).

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A328722 Dirichlet g.f.: 1 / zeta(s-1)^2.

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A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

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A327960 Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1)^2).

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A120630 Dirichlet inverse of A002654.

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A328502 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).

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Mathematica

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A328641 Dirichlet g.f.: zeta(s)^2 / zeta(s-1)^2.

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