Displaying 1-10 of 11 results found.
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
COMMENTS
Row sums = A101035: (1, -3, -5, 1, -9, 15, ...). A127140 * [1, 2, 3, ...] = A055615: (1, -2, -3, 0, -5, 6, -7, ...).
EXAMPLE
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;
...
Dirichlet inverse of the Abelian group count ( A000688).
+10
6
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
COMMENTS
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).
FORMULA
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
EXAMPLE
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.
MAPLE
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:
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;
MATHEMATICA
a[n_] := a[n] = If[n == 1, 1, -Sum[FiniteAbelianGroupCount[n/d] a[d], {d, Most @ Divisors[n]}]];
Dirichlet g.f.: 1 / zeta(s-1)^2.
+10
4
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
COMMENTS
Dirichlet convolution of A055615 with itself.
FORMULA
a(1) = 1; a(n) = -Sum_{d|n, d<n} A038040(n/d) * a(d). 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
MATHEMATICA
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 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021 (PARI)
A007427(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d))); \\ From A007427
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
FORMULA
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
EXAMPLE
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.
MAPLE
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;
MATHEMATICA
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 *)
PROG
(PARI) 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
Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1)^2).
+10
1
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
COMMENTS
Moebius transform applied twice to A101035.
FORMULA
a(1) = 1; a(n) = -Sum_{d|n, d<n} A060640(n/d) * a(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
MATHEMATICA
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 *)
Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).
+10
1
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
FORMULA
a(1) = 1; a(n) = -Sum_{d|n, d<n} A057660(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 fom an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024
MATHEMATICA
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 *)
PROG
(PARI) a(n)={sumdiv(n, d, eulerphi(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019
Dirichlet g.f.: zeta(s)^2 / zeta(s-1)^2.
+10
1
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
COMMENTS
Dirichlet convolution of A023900 with itself. Inverse Moebius transform of A101035.
FORMULA
a(1) = 1; a(n) = -Sum_{d|n, d<n} A029935(n/d) * a(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
MATHEMATICA
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 *)
PROG
(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv(n, d, eulerphi(d) * eulerphi(n/d))))} \\ Andrew Howroyd, Oct 25 2019
Sum of Pillai's arithmetical function ( A018804) and its Dirichlet inverse.
+10
1
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
COMMENTS
No negative terms in range 1 .. 2^20.
Apparently, A030059 gives the positions of all zeros.
PROG
(PARI)
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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
v101035 = DirInverseCorrect(vector(up_to, n, A018804(n)));
G.f. A(x) satisfies: 1 / (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).
+10
1
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
FORMULA
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).
MATHEMATICA
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 - x) = Product_{i>=1, j>=1} A(x^(i*j)).
+10
1
1, -1, 2, 0, 0, 4, -5, 9, -6, 3, 4, -9, 15, -17, 13, -8, 0, 1, -9, 12, -17, 15, -25, 29, -27, 12, -3, -14, 28, -55, 63, -54, 53, -46, 18, 32, -57, 85, -106, 122, -108, 43, 8, -29, 80, -161, 148, -115, 104, -78, 57, 29, -77, 89, -99, 263, -283, 182, -212, 133, 49
FORMULA
G.f. A(x) satisfies: (1 - x) = Product_{k>=1} A(x^k)^ A000005(k). G.f.: Product_{k>=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).
MATHEMATICA
nmax = 60; A007427[n_] := Sum[MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}]; CoefficientList[Series[Product[(1 - x^k)^ A007427[k], {k, 1, nmax}], {x, 0, nmax}], x]
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