A097945 -id:A097945 - cp's OEIS frontend

A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).

1, -1, -2, 2, -4, 2, -6, 0, 6, 4, -10, -4, -12, 6, 8, 0, -16, -6, -18, -8, 12, 10, -22, 0, 20, 12, 0, -12, -28, -8, -30, 0, 20, 16, 24, 12, -36, 18, 24, 0, -40, -12, -42, -20, -24, 22, -46, 0, 42, -20, 32, -24, -52, 0, 40, 0, 36, 28, -58, 16, -60, 30, -36, 0, 48, -20, -66, -32, 44, -24, -70, 0, -72, 36, -40, -36

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative because A055615 and A057521 are.

Convolving this with Euler phi (A000010) produces A349379.

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

Cf. A055615, A057521, A349442 (Dirichlet inverse), A349443 (sum with it).

Cf. also A097945, A349379.

f[p_, e_] := Which[e > 2, 0, e == 2, p^2 - p, e == 1, 1 - p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A055615(n) = (n*moebius(n));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349441(n) = sumdiv(n,d,A057521(n/d)*A055615(d));

a(n) = Sum_{d|n} A057521(n/d) * A055615(d).

Multiplicative with a(p^e) = 1 - p is e = 1, p^2 - p if e = 2, and 0 otherwise. - Amiram Eldar, Nov 19 2021

A097946 a(n) = A008683(n)*A014197(n) where A008683 is the Moebius (or Mobius) function mu(n) and A014197 is the number of numbers m with Euler phi(m) = n.

Original entry on oeis.org

2, -3, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gerald McGarvey, Sep 04 2004

sign

For n < 93 and a(n) not 0, n = p - 1 where p is prime and therefore in A077064 (Squarefree numbers of form prime - 1.)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A008683, A014197, A077064, A097945.

A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ This function from M. F. Hasler, Oct 05 2009
A097946(n) = moebius(n)*A014197(n); \\ Antti Karttunen, Jul 19 2017

A191750 Dirichlet convolution of A000012 with A007947.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 7, 7, 18, 12, 20, 14, 24, 24, 9, 18, 21, 20, 30, 32, 36, 24, 28, 11, 42, 10, 40, 30, 72, 32, 11, 48, 54, 48, 35, 38, 60, 56, 42, 42, 96, 44, 60, 42, 72, 48, 36, 15, 33, 72, 70, 54, 30, 72, 56, 80, 90, 60, 120, 62, 96, 56, 13, 84, 144, 68, 90, 96, 144, 72

Offset: 1

Enrique Pérez Herrero, Jun 22 2011

The squarefree kernel of n is sometimes called rad(n).
Sequence is multiplicative with a(p^e) = 1 + p*e.
Dirichlet convolution of A000005 with the function of absolute values of A097945. - R. J. Mathar, Jul 12 2011
Dirichlet convolution of phi(n)*mu(n)^2 with tau(n). - Richard L. Ollerton, May 07 2021

			The divisors of 12 are 1,2,3,4,6 and 12, the squarefree kernels of these numbers are 1,2,3,2,6 and 6, so a(12) = 1+2+3+2+6+6 = 20.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1837 from Vincenzo Librandi)
Wikipedia, Dirichlet convolution.
Wikipedia, Radical of an integer.

Cf. A007947, A000012 (all 1's sequence), A005117, A073355.

A007947:=func< n | &*PrimeDivisors(n) >; A191750:=func< n | &+[ A007947(d): d in Divisors(n) ] >; [ A191750(n): n in [1..80] ]; // Klaus Brockhaus, Jun 27 2011

with(numtheory): A191750 := n -> add(ilcm(op(factorset(k))),k=divisors(n)):
seq(A191750(i), i=1..80); # Peter Luschny, Jun 23 2011

rad[n_]:=Times@@(FactorInteger[n][[All,1]]); A191750[n_]:=Plus@@rad/@Divisors[n]; Array[A191750,50]
a[1] = 1; a[n_] := Times @@ ((1 + First[#] * Last[#])& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

rad(n)=local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]);
A191750(n)=sumdiv(n, d, rad(d))

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 19 2020

a(n) = Sum_{d|n} rad(d) = Sum_{d|n} A007947(d).
a(n) <= sigma_1(n) = A000203(n); equality holds if n is a squarefree number (A005117).
Dirichlet g.f.: zeta^2(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jul 12 2011
G.f.: Sum_{k>=1} rad(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Sum_{d|n} mu(d)^2*phi(d)*tau(n/d). - Ridouane Oudra, Nov 19 2019
From Vaclav Kotesovec, Jun 19 2020: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).
Dirichlet g.f.: zeta(s)^2 * zeta(s-1) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.70444220099916559... (End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2*tau(gcd(n,k)).
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 5, 4, 2, 8, 2, 4, 4, 16, 2, 10, 2, 8, 4, 4, 2, 16, 7, 4, 14, 8, 2, 8, 2, 32, 4, 4, 4, 20, 2, 4, 4, 16, 2, 8, 2, 8, 10, 4, 2, 32, 9, 14, 4, 8, 2, 28, 4, 16, 4, 4, 2, 16, 2, 4, 10, 64, 4, 8, 2, 8, 4, 8, 2, 40, 2, 4, 14, 8, 4, 8, 2, 32, 41, 4, 2, 16, 4

Offset: 1

Ilya Gutkovskiy, Oct 17 2019

Inverse Moebius transform of A003557.

Dirichlet convolution of A000203 with A097945.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A000010, A000079 (fixed points), A000203, A003557, A007947, A008683, A098108 (parity of a(n)), A191750, A300717, A335032.

Table[Sum[d/Last[Select[Divisors[d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 85}]
Table[Sum[MoebiusMu[n/d] EulerPhi[n/d] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 85}]
f[p_, e_] := 1 + (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

G.f.: Sum_{k>=1} (k / rad(k)) * x^k / (1 - x^k), where rad = A007947.
a(n) = Sum_{d|n} A003557(d).
a(n) = Sum_{d|n} mu(n/d) * phi(n/d) * sigma(d), where mu = A008683, phi = A000010 and sigma = A000203.
a(p) = 2, where p is prime.
From Vaclav Kotesovec, Jun 20 2020: (Start)
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 1/(p^s - p)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) + p^(-s)). (End)
Multiplicative with a(p^e) = 1 + (p^e-1)/(p-1). - Amiram Eldar, Oct 14 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*sigma(gcd(n,k)).
a(n) = Sum_{k=1..n} mu(gcd(n,k))*sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A335032 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(1 - s) - p^(-s)).

Original entry on oeis.org

1, 4, 6, 10, 10, 24, 14, 22, 21, 40, 22, 60, 26, 56, 60, 46, 34, 84, 38, 100, 84, 88, 46, 132, 55, 104, 66, 140, 58, 240, 62, 94, 132, 136, 140, 210, 74, 152, 156, 220, 82, 336, 86, 220, 210, 184, 94, 276, 105, 220, 204, 260, 106, 264, 220, 308, 228, 232, 118

Offset: 1

Vaclav Kotesovec, Jun 20 2020

Dirichlet convolution of A000203 with abs(A097945).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A000203, A001620, A097945, A176345, A326306.

Table[Sum[DivisorSigma[1, n/d] * Abs[MoebiusMu[d]] * EulerPhi[d], {d, Divisors[n]}], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)/(1 - p*X))[n], ", "))

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X/(1 - X))/(1 - p*X))[n], ", "))

Dirichlet g.f.: zeta(s) * zeta(s-1)^2 / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).
Dirichlet g.f.: zeta(s) * zeta(s-1)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).
Let f(s) = Product_{primes p} (1 - 1/(p^s + p)), then Sum_{k=1..n} a(k) ~ n^2 * ((log(n)/2 + gamma - 3*zeta'(2)/Pi^2 - 1/4)*f(2) + f'(2)/2), where f(2) = A065463 = Product_{primes p} (1 - 1/(p*(p+1))) = 0.7044422009991655927366033503266372..., f'(2) = f(2) * Sum_{primes p} p*log(p) / ((p+1)*(p^2+p-1)) = 0.23219454323726621271960146689644280341444084188447499043209938838191022838..., for zeta'(2) see A073002 and gamma is the Euler-Mascheroni constant A001620.
a(n) = Sum_{d|n} A176345(d). - Ridouane Oudra, Jan 14 2022
Multiplicative with a(p^e) = sigma(p^e) + p^e - 1. - Amiram Eldar, Dec 25 2022

A341635 a(n) = Sum_{d|n} phi(d) * mu(d) * mu(n/d).

Original entry on oeis.org

1, -2, -3, 1, -5, 6, -7, 0, 2, 10, -11, -3, -13, 14, 15, 0, -17, -4, -19, -5, 21, 22, -23, 0, 4, 26, 0, -7, -29, -30, -31, 0, 33, 34, 35, 2, -37, 38, 39, 0, -41, -42, -43, -11, -10, 46, -47, 0, 6, -8, 51, -13, -53, 0, 55, 0, 57, 58, -59, 15, -61, 62, -14, 0, 65

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Dirichlet inverse of A003967.
Moebius transform of A097945.
From Vaclav Kotesovec, Feb 19 2021: (Start)
Abs(a(n)) <= n.
a(n) = n iff n is in A030229. (End)

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

Cf. A000010, A003967, A007427, A007431, A008683, A030229 (fixed points), A046099 (positions of 0's), A068341, A097945, A276833.

Table[Sum[EulerPhi[d] MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 65}]
Table[Sum[MoebiusMu[GCD[n, k]] MoebiusMu[n/GCD[n, k]], {k, n}], {n, 65}]

a(n) = sumdiv(n, d, eulerphi(d)*moebius(d)*moebius(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} mu(gcd(n,k)) * mu(n/gcd(n,k)).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A003967(n/d) * a(d).
a(n) = Sum_{d|n} mu(n/d) * A097945(d).
Multiplicative with a(p^e) = -p if e=1, p-1 if e=2, and 0 otherwise. - Amiram Eldar, Feb 19 2021

A349911 Dirichlet inverse of A336466, which is fully multiplicative with a(p) = oddpart(p-1).

Original entry on oeis.org

1, -1, -1, 0, -1, 1, -3, 0, 0, 1, -5, 0, -3, 3, 1, 0, -1, 0, -9, 0, 3, 5, -11, 0, 0, 3, 0, 0, -7, -1, -15, 0, 5, 1, 3, 0, -9, 9, 3, 0, -5, -3, -21, 0, 0, 11, -23, 0, 0, 0, 1, 0, -13, 0, 5, 0, 9, 7, -29, 0, -15, 15, 0, 0, 3, -5, -33, 0, 11, -3, -35, 0, -9, 9, 0, 0, 15, -3, -39, 0, 0, 5, -41, 0, 1, 21, 7, 0, -11, 0

Offset: 1

Antti Karttunen, Dec 08 2021

Multiplicative because A336466 is.

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

Cf. A336466, A349912.

Cf. also A097945.

f[p_, e_] := ((p-1)/2^IntegerExponent[p-1, 2])^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,A000265(f[k,1]-1)^f[k,2]); };
memoA349911 = Map();
A349911(n) = if(1==n,1,my(v); if(mapisdefined(memoA349911,n,&v), v, v = -sumdiv(n,d,if(dA336466(n/d)*A349911(d),0)); mapput(memoA349911,n,v); (v)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A336466(n/d) * a(d).

a(n) = A349912(n) - A336466(n).

A054585 Sum_{d=1..n} phi(d)*mu(d).

Original entry on oeis.org

0, 1, 0, -2, -2, -6, -4, -10, -10, -10, -6, -16, -16, -28, -22, -14, -14, -30, -30, -48, -48, -36, -26, -48, -48, -48, -36, -36, -36, -64, -72, -102, -102, -82, -66, -42, -42, -78, -60, -36, -36, -76, -88, -130, -130, -130, -108, -154, -154, -154, -154, -122, -122, -174, -174, -134, -134, -98, -70, -128

Offset: 0

N. J. A. Sloane, Apr 12 2000

sign

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Partial sums of A097945.

a(n)=my(s); forsquarefree(k=1, n, s += moebius(k)*eulerphi(k)); s \\ Charles R Greathouse IV, Jan 07 2018

first(n)=my(v=vector(n),s); forsquarefree(k=1, n, v[k[1]] = s+=moebius(k)*eulerphi(k)); for(k=4,n, if(v[k]==0, v[k]=v[k-1])); concat(0, v) \\ Charles R Greathouse IV, Jan 07 2018

A143153 Triangle read by rows, A054525 * (A020639 * 0^(n-k)), 1<=k<=n.

Original entry on oeis.org

1, -1, 2, -1, 0, 3, 0, -2, 0, 2, -1, 0, 0, 0, 5, 1, -2, -3, 0, 0, 2, -1, 0, 0, 0, 0, 0, 7, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -3, 0, 0, 0, 0, 0, 3, 1, -2, 0, 0, -5, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gary W. Adamson & Mats Granvik, Jul 27 2008

Right border = A020639, Lpf(n): (1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11,...).
Left border = mu(n), A008683: (1, -1, -1, 0, -1, 1, -1,...).
Row sums = a signed version of A097945: (1, -1, -2, 0, -4, 2, -6,...) such that parity = (+) iff mu(n) = +.

			First few rows of the triangle =
1;
-1, 2;
-1, 0, 3;
0, -2, 0, 2;
-1, 0, 0, 0, 5;
1, -2, -3, 0, 0, 2;
-1, 0, 0, 0, 0, 0, 7;
0, 0, 0, -2, 0, 0, 0, 2;
0, 0, -3, 0, 0, 0, 0, 0, 3;
1, -2, 0, 0, -5, 0, 0, 0, 0, 2;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 2;
...

Cf. A008683, A097945, A020639, A054525.

Triangle read by rows, A054525 * (A020639 * 0^(n-k)), 1<=k<=n; where A020639 = Lpf(n): (1, 2, 3, 2, 5, 2, 7, 2, 3, 2,...) and A054525 = the Mobius transform.

A265204 Sum of phi(i) over squarefree numbers i <= n.

Original entry on oeis.org

1, 2, 4, 4, 8, 10, 16, 16, 16, 20, 30, 30, 42, 48, 56, 56, 72, 72, 90, 90, 102, 112, 134, 134, 134, 146, 146, 146, 174, 182, 212, 212, 232, 248, 272, 272, 308, 326, 350, 350, 390, 402, 444, 444, 444, 466, 512, 512, 512, 512, 544, 544, 596, 596, 636, 636, 672, 700, 758, 758, 818, 848, 848, 848, 896, 916, 982, 982, 1026, 1050

Offset: 1

Jeffrey Shallit, Dec 04 2015

nonn

Partial sums of absolute values of A097945. - Robert Israel, Dec 10 2015

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A097945.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1))+
      `if`(issqrfree(n), phi(n), 0)
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Dec 04 2015
N:= 1000: # to get a(1) to a(N)
V:= Vector(N, 1):
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
for p in Primes do
  J1:= [seq(i,i=p..N,p)];
  J2:= [seq(i,i=p^2..N,p^2)];
  V[J1]:= V[J1] * (p-1);
  V[J2]:= 0;
od:
ListTools[PartialSums](convert(V,list)); # Robert Israel, Dec 10 2015

Table[Sum[EulerPhi@ i, {i, Select[Range@ n, SquareFreeQ]}], {n, 70}] (* Michael De Vlieger, Dec 10 2015 *)

a(n) = sum(i=1, n, eulerphi(i)*issquarefree(i)) \\ Anders Hellström, Dec 04 2015

use ntheory ":all"; sub an { vecsum(map { is_square_free($) ? euler_phi($) : () } 1..shift); } say an($) for 1..70; # _Dana Jacobsen, Dec 10 2015

cp's OEIS Frontend

A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A097946 a(n) = A008683(n)*A014197(n) where A008683 is the Moebius (or Mobius) function mu(n) and A014197 is the number of numbers m with Euler phi(m) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A191750 Dirichlet convolution of A000012 with A007947.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A335032 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(1 - s) - p^(-s)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A341635 a(n) = Sum_{d|n} phi(d) * mu(d) * mu(n/d).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349911 Dirichlet inverse of A336466, which is fully multiplicative with a(p) = oddpart(p-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI