A007427 -id:A007427 - cp's OEIS frontend

A001159 sigma_4(n): sum of 4th powers of divisors of n.

1, 17, 82, 273, 626, 1394, 2402, 4369, 6643, 10642, 14642, 22386, 28562, 40834, 51332, 69905, 83522, 112931, 130322, 170898, 196964, 248914, 279842, 358258, 391251, 485554, 538084, 655746, 707282, 872644, 923522, 1118481, 1200644

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001157, A001158.

[DivisorSigma(4,n): n in [1..40]]; // Bruno Berselli, Apr 10 2013

with(numtheory); A001159 := proc(n) sigma[4](n) ; end proc: # R. J. Mathar, Feb 04 2011

lst={}; Do[AppendTo[lst, DivisorSigma[4,n]], {n,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[4,Range[40]] (* Harvey P. Dale, Apr 28 2013 *)

makelist(divsum(n,4),n,1,100); /* Emanuele Munarini, Mar 26 2011 */

N=99;q='q+O('q^N);
Vec(sum(n=1,N,n^4*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */

[sigma(n,4)for n in range(1,34)] # Zerinvary Lajos_, Jun 04 2009

Multiplicative with a(p^e) = (p^(4e+4)-1)/(p^4-1). - David W. Wilson, Aug 01 2001
G.f. Sum_{k>=1} k^4*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{j>=1} (1-x^j)^(j^3)) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Dirichlet g.f.: zeta(s)*zeta(s-4). - R. J. Mathar, Feb 04 2011
a(n) = Sum_{d|n} tau_{-2}^(d)*J_4(n/d), where tau_{-2} is A007427 and J_4 A059377. - Enrique Pérez Herrero, Jan 19 2013
G..f.: Sum_{n >= 1} A(4,x^n)/(1 - x^n)^5, where A(4,x) = x + 11*x^2 + 11*x^3 + x^4 is the 4th Eulerian polynomial - see A008292. - Peter Bala, Jan 11 2021
a(n) = Sum_{1 <= i, j, k, l <= n} tau(gcd(i, j, k, l, n)) = Sum_{d divides n} tau(d) * J_4(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024

A007431 a(n) = Sum_{d|n} phi(d)*mu(n/d).

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 0, 5, 2, 4, 0, 9, 1, 11, 0, 3, 4, 15, 0, 17, 3, 5, 0, 21, 2, 16, 0, 12, 5, 27, 0, 29, 8, 9, 0, 15, 4, 35, 0, 11, 6, 39, 0, 41, 9, 12, 0, 45, 4, 36, 0, 15, 11, 51, 0, 27, 10, 17, 0, 57, 3, 59, 0, 20, 16, 33, 0, 65, 15, 21, 0, 69, 8, 71, 0, 16, 17, 45, 0, 77, 12, 36, 0, 81, 5, 45, 0

Offset: 0

N. J. A. Sloane

Also Moebius transform applied twice to natural numbers.
Also number of complex primitive Dirichlet characters modulo n and Sum_{k=1..n} a(k) is asymptotic to (18/Pi^4)*n^2. - Steven Finch, Feb 16 2006
Dirichlet convolution of phi(n) and mu(n). - Richard L. Ollerton, May 07 2021
From Jianing Song, May 21 2022: (Start)
a(n) is the number of degree-psi(n) primitive Dirichlet characters mod n, where psi = A002322. Also, a(n) is the number of degree-(k*psi(n)) primitive Dirichlet characters mod n for all k >= 1.
a(n) is the maximum element in the n-th row of A354058 (or A354061). (End)

			From _Jianing Song_, May 21 2022: (Start)
a(45) = 12: psi(45) = 12, there are 3 degree-12 primitive characters modulo 5 and 4 degree-12 primitive characters modulo 9, so a(45) = 3 * 4 = 12.
a(63) = 20: psi(63) = 6, there are 5 sextic primitive characters modulo 7 and 4 sextic primitive characters modulo 9, so a(63) = 5 * 4 = 20. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 0..20000 (terms 0..1000 from T. D. Noe)
H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973) 452-455.
Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.
N. J. A. Sloane, Transforms

Cf. A007432.

Cf. A000010, A008683, A130054 (Dirichlet inverse), A354058, A354061.

a007431 0 = 0
a007431 n = sum $ map (a008683 . gcd n) [1..n]
-- Reinhard Zumkeller, Jan 06 2014

[0] cat [&+[EulerPhi(d)*MoebiusMu(Floor(n/d)):d in Divisors(n)]:n in [1..90]]; // Marius A. Burtea, Aug 10 2019

with(numtheory); f:=n->add( phi(d)*mobius(n/d), d in divisors(n)); [seq(f(n),n=0..120)];

Table[Sum[EulerPhi[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 0, 86}] (* Jean-François Alcover, Apr 04 2011 *)
Table[DirichletConvolve[MoebiusMu[n], EulerPhi[n], n, m], {m, 86}] (* Jan Mangaldan, Mar 15 2013 *)
f[p_, e_] := If[e == 1, p-2, p^e - 2*p^(e-1) + p^(e-2)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 23 2020 *)

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

a(n) = sumdiv(n,d, moebius(d) * eulerphi(n/d) ); \\ Joerg Arndt, Apr 14 2013

A007431(n) = if(!n,n,my(f=factor(n)); prod(i=1, #f~, if(1==f[i, 2], f[i, 1]-2, ((f[i,1]-1)^2)*(f[i, 1]^(f[i, 2]-2))))); \\ Antti Karttunen, Dec 15 2024, after Vladeta Jovovic's multiplicative formula

Multiplicative with a(p) = p-2 and a(p^e) = (p-1)^2*p^(e-2) for e > 1. - Vladeta Jovovic, Jan 25 2002
Dirichlet g.f.: zeta(s-1)/zeta^2(s).
a(n) = Sum_{k=1..n} mu(gcd(n,k)) for n > 0. - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{k=1..n} (phi(gcd(k,n)) * cos(2*Pi*k/n)). - Enrique Pérez Herrero, Jan 18 2013
a(n) = Sum_{d|n} tau_{-2}(d)*n/d = Sum_{d|n} tau_{-3}(d)*sigma_1(n/d), where tau_{-3} is A007428, tau_{-2} A007427 and sigma_1 A000203. - Enrique Pérez Herrero, Jan 19 2013
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} mu(n)*x^n/(1 - x^n)^2. - Ilya Gutkovskiy, Apr 25 2017
Sum_{k=1..n} a(k) ~ 18 * n^2 / Pi^4. - Vaclav Kotesovec, Nov 04 2018
Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} phi(n)*x^n. - Mamuka Jibladze, Aug 09 2019
Sum_{d|n} a(d) = phi(n) (A000010). - Amiram Eldar, Jun 23 2020
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = A354058(n,psi(n)) = A354061(n,psi(n)) with psi = A002322. - Jianing Song, May 21 2022

A046692 Dirichlet inverse of sigma function (A000203).

Original entry on oeis.org

1, -3, -4, 2, -6, 12, -8, 0, 3, 18, -12, -8, -14, 24, 24, 0, -18, -9, -20, -12, 32, 36, -24, 0, 5, 42, 0, -16, -30, -72, -32, 0, 48, 54, 48, 6, -38, 60, 56, 0, -42, -96, -44, -24, -18, 72, -48, 0, 7, -15, 72, -28, -54, 0, 72, 0, 80, 90, -60, 48, -62, 96, -24, 0, 84, -144, -68, -36, 96, -144, -72, 0, -74, 114, -20, -40, 96, -168

Offset: 1

Andrew R. Feist (arf22540(AT)cmsu2.cmsu.edu)

			a(36) = a(2^2*3^2) = 2*3 = 6.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
Andrew R. Feist, Fun With the Sigma-Function, unpub.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Cf. A000203, A053822, A053825, A053826, A178448.

t := 1; a := proc(n,t) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 then t1 := t1+d^t*mobius(d)*mobius(n/d); fi; od; t1; end;

a[n_] := (k = 0; Do[If[Mod[n, d] == 0, k = k + d*MoebiusMu[d]*MoebiusMu[n/d]], {d, 1, n}]; k); Table[a[n], {n, 1, 78}](* Jean-François Alcover, Oct 13 2011, after Maple *)
f[p_, e_] := Which[e == 1, -p-1, e == 2, p, e >= 3, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

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

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n)))} \\ Andrew Howroyd, Aug 05 2018

a(p) = -p-1, a(p^2) = p, a(p^k) = 0 for k > 2.
Dirichlet g.f.: 1/(zeta(s)*zeta(s-1)). - Benedict W. J. Irwin, Jul 10 2018
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} sigma(k)*A(x^k). - Ilya Gutkovskiy, May 11 2019
From Peter Bala, Jan 17 2024: (Start)
a(n) = Sum_{d divides n} d*mu(d)*mu(n/d). See Brown, p. 408.
a(n) = - Sum_{d divides n, d < n} a(d)*sigma_1(n/d).
a(n) = Sum_{d divides n} d*a(d)*J_2(n/d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{d divides n} d*A007427(d)*phi(n/d), where A007427 is the Dirichlet inverse of the tau function.
More generally, a(n) = Sum_{d divides n} d*sigma_[r]^(-1)(d)*J_(r+1)(n/d), where sigma_[r]^(-1) denotes the Dirichlet inverse of the function sigma_[r] = Sum_{d divides n} d^r.
a(n) = Sum_{k = 1..n} gcd(k, n)*A007427(gcd(k, n)).
a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)*a(gcd(j, k, n)). (End)
Sum_{k=1..n} abs(a(k)) ~ 45*n^2/Pi^4. - Vaclav Kotesovec, May 30 2024

Corrected by T. D. Noe, Nov 13 2006

A296302 Number of aperiodic compositions of n with relatively prime parts. Number of compositions of n with relatively prime parts and relatively prime run-lengths.

Original entry on oeis.org

1, 0, 2, 5, 14, 24, 62, 114, 249, 480, 1022, 1978, 4094, 8064, 16348, 32520, 65534, 130512, 262142, 523270, 1048444, 2095104, 4194302, 8384316, 16777185, 33546240, 67108356, 134201398, 268435454, 536837136, 1073741822, 2147418240, 4294965244, 8589803520

Offset: 1

Gus Wiseman, Dec 11 2017

nonn

			The a(6) = 24 aperiodic compositions with relatively prime parts are:
(15), (51),
(114), (123), (132), (141), (213), (231), (312), (321), (411),
(1113), (1122), (1131), (1221), (1311), (2112), (2211), (3111),
(11112), (11121), (11211), (12111), (21111).

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000005, A000740, A000837, A007427, A008683, A008965, A059966, A060223, A100953, A228369, A281013.

Table[DivisorSum[n,Function[d,MoebiusMu[n/d]*DivisorSum[d,MoebiusMu[#]*2^(d/#-1)&]]],{n,20}]

a = mu * mu * c, where * is Dirichlet convolution and c(n) = 2^(n-1).

A007428 Moebius transform applied thrice to sequence 1,0,0,0,....

Original entry on oeis.org

1, -3, -3, 3, -3, 9, -3, -1, 3, 9, -3, -9, -3, 9, 9, 0, -3, -9, -3, -9, 9, 9, -3, 3, 3, 9, -1, -9, -3, -27, -3, 0, 9, 9, 9, 9, -3, 9, 9, 3, -3, -27, -3, -9, -9, 9, -3, 0, 3, -9, 9, -9, -3, 3, 9, 3, 9, 9, -3, 27, -3, 9, -9, 0, 9, -27, -3, -9, 9, -27, -3, -3, -3, 9, -9, -9, 9, -27

Offset: 1

N. J. A. Sloane

Dirichlet inverse of A007425. - R. J. Mathar, Jul 15 2010

abs(a(n)) is the number of ways to write n=xyz where x,y,z are squarefree numbers. - Benoit Cloitre, Jan 02 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Consecutive nested Dirichlet convolution: A063524, A008683 or A007427. - Enrique Pérez Herrero, Jul 12 2010

Cf. A124010.

a007428 n = product
   [a007318' 3 e * cycle [1,-1] !! fromIntegral e | e <- a124010_row n]
-- Reinhard Zumkeller, Oct 09 2013

möbius := proc(a)  local b, i, mo: b := NULL:
mo := (m,n) -> `if`(irem(m,n) = 0, numtheory:-mobius(m/n), 0);
for i to nops(a) do b := b, add(mo(i,j)*a[j], j=1..i) od: [b] end:
(möbius@@3)([1, seq(0, i=1..77)]); # Peter Luschny, Sep 08 2017

tau[1,n_Integer]:=1; SetAttributes[tau, Listable];
tau[k_Integer,n_Integer]:=Plus@@(tau[k-1,Divisors[n]])/; k > 1;
tau[k_Integer,n_Integer]:=Plus@@(tau[k+1,Divisors[n]]*MoebiusMu[n/Divisors[n]]); k<1;
A007428[n_]:=tau[ -3,n]; (* Enrique Pérez Herrero, Jul 12 2010 *)
a[n_] := Which[n==1, 1, PrimeQ[n], -3, True, Times @@ Map[Function[e, Binomial[3, e] (-1)^e], FactorInteger[n][[All, 2]]]];
Array[a, 100] (* Jean-François Alcover, Jun 20 2018 *)

a(n) = {my(f=factor(n)); for (k=1, #f~, e = f[k,2]; f[k,1] = binomial(3, e)*(-1)^e; f[k,2] = 1); factorback(f);} \\ Michel Marcus, Jan 03 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (3 choose e) (-1)^e.
Dirichlet g.f.: 1/zeta(s)^3.
From Enrique Pérez Herrero, Jul 12 2010: (Start)
a(n^3) = A008683(n).
a(s) = (-3)^A001221(s) provided s is a squarefree number (A005117). (End)
a(A046101(n)) = 0. - Enrique Pérez Herrero, Sep 07 2017
a(n) = Sum_{a*b*c=n} mu(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

A077049 Left summatory matrix, T, by antidiagonals upwards.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Oct 22 2002

If S = (s(1), s(2), ...) is a sequence written as a column vector, then T*S is the summatory sequence of S; i.e., its n-th term is Sum_{k|n} s(k). T is the inverse of the left Moebius transformation matrix, A077050. Except for the first term in some cases, column 1 of T^(-2) is A007427, column 1 of T^(-1) is A008683, Column c of T^2 is A000005, column 1 of T^3 is A007425.
This is essentially the same as A051731, which includes only the triangle. Note that the standard in the OEIS is left to right antidiagonals, which would make this the right summatory matrix, and A077051 the left one. - Franklin T. Adams-Watters, Apr 08 2009
From Gary W. Adamson, Apr 28 2010: (Start)
As defined with antidiagonals of the array = the triangle shown in the example section. Row sums of this triangle = A032741 (with a different offset): 1, 1, 2, 1, 3, 1, 3, ...
Let the triangle = M. Then lim_{n->inf} M^n = A002033, the left-shifted vector considered as a sequence: (1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, ...). (End)

			T(4,2) = 1 since 2 divides 4. Northwest corner:
  1 0 0 0 0 0
  1 1 0 0 0 0
  1 0 1 0 0 0
  1 1 0 1 0 0
  1 0 0 0 1 0
  1 1 1 0 0 1
From _Gary W. Adamson_, Apr 28 2010: (Start)
First few rows of the triangle (when T is read by antidiagonals upwards):
  1;
  1, 0;
  1, 1, 0;
  1, 0, 0, 0;
  1, 1, 1, 0, 0;
  1, 0, 0, 0, 0, 0;
  1, 1, 0, 1, 0, 0, 0;
  1, 0, 1, 0, 0, 0, 0, 0;
  1, 1, 0, 0, 1, 0, 0, 0, 0;
  ... (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows 1 <= n <= 150).
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

Cf. A051731, A077050, A077051, A077052, A000005 (row sums).

Cf. A032741, A002033. - Gary W. Adamson, Apr 28 2010

A077049 := proc(n,k)
    if modp(n,k) = 0 then
        1;
    else
        0 ;
    end if;
end proc:
for d from 2 to 10 do
    for k from 1 to d-1 do
        n := d-k ;
        printf("%d,",A077049(n,k)) ;
    end do:
end do: # R. J. Mathar, Jul 22 2017

With[{nn = 14}, DeleteCases[#, -1] & /@ Transpose@ Table[Take[#, nn] &@ Flatten@ Join[ConstantArray[-1, k - 1], ConstantArray[Reverse@ IntegerDigits[2^(k - 1), 2], Ceiling[(nn - k + 1)/k]]], {k, nn}]] // Flatten (* Michael De Vlieger, Jul 22 2017 *)

nn=10; matrix(nn, nn, n, k, if (n % k, 0, 1)) \\ Michel Marcus, May 21 2015

def T(n, k):
    return 1 if n%k==0 else 0
for n in range(1, 11): print([T(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jul 22 2017

T(n,k)=1 if k|n, otherwise T(n,k)=0, k >= 1, n >= 1.
From Boris Putievskiy, May 08 2013: (Start)
As table T(n,k) = floor(k/n) - floor((k-1)/n).
As linear sequence a(n) = floor(A004736(n)/A002260(n)) - floor((A004736(n)-1)/A002260(n)); a(n) = floor(j/i)-floor((j-1)/i), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

Name edited by Petros Hadjicostas, Jul 27 2019

A101035 Dirichlet inverse of the gcd-sum function (A018804).

Original entry on oeis.org

1, -3, -5, 1, -9, 15, -13, 1, 4, 27, -21, -5, -25, 39, 45, 1, -33, -12, -37, -9, 65, 63, -45, -5, 16, 75, 4, -13, -57, -135, -61, 1, 105, 99, 117, 4, -73, 111, 125, -9, -81, -195, -85, -21, -36, 135, -93, -5, 36, -48, 165, -25, -105, -12, 189, -13, 185, 171, -117, 45, -121, 183, -52, 1, 225, -315, -133, -33, 225, -351, -141, 4

Offset: 1

Gerard P. Michon, Nov 27 2004

			a(4)=1, a(8)=1, a(16)=1, a(32)=1, etc. because of the multiplicative definition for powers of 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. P. Michon, Multiplicative Functions.

Cf. A018804, A055615, A046692, A023900, A007427, A053822, A053825, A053826.

Cf. A008683.

a101035 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p 1 = 1 - 2 * p
   f p e = (p - 1) ^ 2
-- Reinhard Zumkeller, Jul 16 2012

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[ f[n/d]*DirichletInverse[f][d], {d, Most[ Divisors[n]]}]; GCDSum[n_] := Sum[ GCD[n, k], {k, 1, n}]; Table[ DirichletInverse[ GCDSum][n], {n, 1, 72}](* Jean-François Alcover, Dec 12 2011 *)
f[p_, e_] := If[e == 1, 1 - 2*p, (p - 1)^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 06 2022 *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv(n, d, n*eulerphi(d)/d)))} \\ Andrew Howroyd, Aug 05 2018

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

Multiplicative function with a(p)=1-2p and a(p^e)=(p-1)^2 when e>1 [p prime].
Dirichlet g.f.: zeta(s)/zeta^2(s-1). - R. J. Mathar, Apr 10 2011
a(n) = Sum{d|n} tau_{-2}(d)*d, where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
Conjecture: Logarithmic g.f. Sum_{n>0,k>0} mu(n)*mu(k)*log(1/(1-x^(n*k))). - Benedict W. J. Irwin, Jul 26 2017

A226602 Number of ordered triples (i,j,k) with ijk = n, i,j,k >= 0 and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 1, 3, 3, 6, 3, 9, 3, 9, 6, 9, 3, 18, 3, 9, 9, 12, 3, 18, 3, 18, 9, 9, 3, 27, 6, 9, 9, 18, 3, 27, 3, 15, 9, 9, 9, 36, 3, 9, 9, 27, 3, 27, 3, 18, 18, 9, 3, 36, 6, 18, 9, 18, 3, 27, 9, 27, 9, 9, 3, 54, 3, 9, 18, 18, 9, 27, 3, 18, 9, 27, 3, 54, 3, 9, 18, 18

Offset: 0

Robert Price, Jun 13 2013

Note that gcd(0,m) = m for any m.

a(n) is the number of cubefree divisors summed over the divisors of n. In other words, a(n) = Sum_{d|n} A073184(d). - Geoffrey Critzer, Mar 20 2015

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 101 terms from Robert Price)

Cf. A000005, A007427, A008836, A034444, A056624, A073184, A092520, A100450, A212793, A226357, A226359.

with(numtheory):
b:= proc(n, t, g) option remember; `if`(t=0,
      `if`(igcd(n, g)=1, 1, 0), add(b(n/d, t-1,
      igcd(g, d)), d=divisors(n)))
    end:
a:= n-> `if`(n=0, 1, b(n, 2, 0)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 20 2015

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n && GCD[i, j, k] <= 1, {i, j, k}], {i, 0,n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]
a[0] = a[1] = 1; a[n_] := Times @@ (3 * Last[#] & /@ FactorInteger[n]); Array[a, 100, 0] (* Amiram Eldar, Sep 14 2020 *)

from math import prod
from sympy import factorint
def A226602(n): return prod(3*e for e in factorint(n).values()) if n else 1 # Chai Wah Wu, Dec 26 2022

From Geoffrey Critzer, Mar 20 2015: (Start)
If n = p_1^e_1*p_2^e_2*...*p_r^e_r then a(n) = Product_{i=1..r} 3*e_i.
Dirichlet g.f.: zeta(s)^3/zeta(3*s). (End)
From Werner Schulte, May 13 2018: (Start)
Multiplicative with a(p^e) = 3*e, p prime and e>0.
Dirichlet inverse b(n), n>0, is multiplicative with b(1) = 1, and for p prime and e>0: b(p^e)=0 if e mod 3 = 0 otherwise b(p^e)=3*(-1)^(e mod 3).
Dirichlet convolution with A007427(n) yields A212793(n).
Dirichlet convolution with A008836(n) yields A092520(n).
Equals Dirichlet convolution of A034444(n) and A056624(n).
Equals Dirichlet convolution of A000005(n) and A212793(n). (End)
Sum_{k=1..n} a(k) ~ n/(2*Zeta(3)) * (log(n)^2 + 2*log(n) * (-1 + 3*gamma - 3*Zeta'(3)/Zeta(3)) + 2 + 6*gamma^2 - 6*sg1 + 6*Zeta'(3)/Zeta(3) + 18*Zeta'(3)^2/Zeta(3)^2 - 6*gamma*(1 + 3*Zeta'(3)/Zeta(3)) - 9*Zeta''(3)/Zeta(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 07 2019
a(n) = A005361(n) * A074816(n). - Vaclav Kotesovec, Feb 27 2023

A227291 Characteristic function of squarefree numbers squared (A062503).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Ralf Stephan, Jul 05 2013

			a(3) = 0 because 3 is not the square of a squarefree number.
a(4) = 1 because sqrt(4) = 2, a squarefree number.

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A000196, A007427, A008836, A008683, A008966, A010052, A027750, A037213, A225546, A225569, A225817, A307430, A322327, A355448.

Absolute values of A271102.

a227291 n = fromEnum $ (sum $ zipWith (*) mds (reverse mds)) == 1
   where mds = a225817_row n
-- Reinhard Zumkeller, Jul 30 2013, Jul 07 2013

A227291 := proc(n)
    local pe;
    if n = 0 then
        1;
    else
        for pe in ifactors(n)[2] do
            if op(2,pe) <> 2 then
                return 0 ;
            end if;
        end do:
    end if;
    1 ;
end proc:
seq(A227291(n),n=1..100) ; # R. J. Mathar, Feb 07 2023

Table[Abs[Sum[MoebiusMu[n/d], {d,Select[Divisors[n], SquareFreeQ[#] &]}]], {n, 1, 200}] (* Geoffrey Critzer, Mar 18 2015 *)
Module[{nn=120,len,sfr},len=Ceiling[Sqrt[nn]];While[!SquareFreeQ[len],len++];sfr=(Select[Range[len],SquareFreeQ])^2; Table[If[MemberQ[ sfr,n],1,0],{n,nn}]] (* Harvey P. Dale, Nov 27 2024 *)

a(n)=if(n<1, 0, direuler(p=2, n, 1+X^2)[n])

A227291(n) = factorback(apply(e->(2==e), factor(n)[,2])); \\ Antti Karttunen, Jul 14 2022

A227291(n) = (issquare(n) && issquarefree(sqrtint(n))); \\ Antti Karttunen, Jul 14 2022

(define (A227291 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) 1 0) (A227291 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

Dirichlet g.f.: zeta(2s)/zeta(4s) = prod[prime p: 1+p^(-2s) ], see A008966.
a(n) = A008966(A037213(n)), when assumed A008966(0) = 0. - Reinhard Zumkeller, Jul 07 2013
a(n) = A063524(sum(A225817(n,k)*A225817(n,A000005(n)+1-k): k=1..A000005(n))). - Reinhard Zumkeller, Aug 01 2013
Multiplicative with a(p^e) = 1 if e=2, a(p^e) = 0 if e=1 or e>2. - Antti Karttunen, Jul 28 2017
Sum_{k=1..n} a(k) ~ 6*sqrt(n) / Pi^2. - Vaclav Kotesovec, Feb 02 2019
a(n) = A225569(A225546(n)-1). - Peter Munn, Oct 31 2019
From Antti Karttunen, Jul 18 2022: (Start)
a(n) = A010052(n) * A008966(A000196(n)).
a(n) = Sum_{d|n} A008836(n/d) * A307430(d).
a(n) = Sum_{d|n} A007427(n/d) * A322327(d).
(End)

A322328 a(n) = A005361(n) * 4^A001221(n) for n > 0.

Original entry on oeis.org

1, 4, 4, 8, 4, 16, 4, 12, 8, 16, 4, 32, 4, 16, 16, 16, 4, 32, 4, 32, 16, 16, 4, 48, 8, 16, 12, 32, 4, 64, 4, 20, 16, 16, 16, 64, 4, 16, 16, 48, 4, 64, 4, 32, 32, 16, 4, 64, 8, 32, 16, 32, 4, 48, 16, 48, 16, 16, 4, 128, 4, 16, 32, 24, 16, 64, 4, 32, 16, 64, 4

Offset: 1

Werner Schulte, Dec 03 2018

Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see A322327, for k = 3 see A226602 (offset 1), and for k = 4 see this sequence.

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

Cf. A000005, A000007, A001221, A005361, A007427, A008836, A034444, A226602, A322327.

f:= n -> mul(4*t[2],t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Dec 07 2018

a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 4^Length[f] * Times@@f[[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)

a(n) = my(f=factor(n)); vecprod(f[,2])*4^omega(n); \\ Michel Marcus, Dec 04 2018

from math import prod
from sympy import factorint
def A322328(n): return prod(e<<2 for e in factorint(n).values()) # Chai Wah Wu, Dec 24 2022

Multiplicative with a(p^e) = 4*e for prime p and e > 0.
Dirichlet g.f.: (zeta(s))^4 / (zeta(2*s))^2.
Dirichlet inverse is b(n) = a(n) * A008836(n) for n > 0, and b(n) is multiplicative with b(p^e) = 4*e*(-1)^e for prime p and e > 0.
Equals Dirichlet convolution of A034444 with itself.
Equals Dirichlet convolution of A000005 with abs(A007427).

cp's OEIS Frontend

A001159 sigma_4(n): sum of 4th powers of divisors of n.

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A007431 a(n) = Sum_{d|n} phi(d)*mu(n/d).

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A046692 Dirichlet inverse of sigma function (A000203).

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A296302 Number of aperiodic compositions of n with relatively prime parts. Number of compositions of n with relatively prime parts and relatively prime run-lengths.

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A007428 Moebius transform applied thrice to sequence 1,0,0,0,....

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A077049 Left summatory matrix, T, by antidiagonals upwards.

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A226602 Number of ordered triples (i,j,k) with ijk = n, i,j,k >= 0 and gcd(i,j,k) <= 1.