A063441 -id:A063441 - cp's OEIS frontend

A349355 Dirichlet convolution of A003958 with A063441 (Dirichlet inverse of A003959), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

1, -2, -2, -2, -2, 4, -2, -2, -4, 4, -2, 4, -2, 4, 4, -2, -2, 8, -2, 4, 4, 4, -2, 4, -8, 4, -8, 4, -2, -8, -2, -2, 4, 4, 4, 8, -2, 4, 4, 4, -2, -8, -2, 4, 8, 4, -2, 4, -12, 16, 4, 4, -2, 16, 4, 4, 4, 4, -2, -8, -2, 4, 8, -2, 4, -8, -2, 4, 4, -8, -2, 8, -2, 4, 16, 4, 4, -8, -2, 4, -16, 4, -2, -8, 4, 4, 4, 4, -2, -16

Offset: 1

Antti Karttunen, Nov 16 2021

Multiplicative because both A003958 and A063441 are.

In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003959 with factor A003958. For example, convolving this with A003968 (the Möbius transform of A003959) produces A003966, the Möbius transform of A003958.

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

Cf. A003958, A003959, A003966, A003968, A063441, A349356 (Dirichlet inverse), A349357 (sum with it).

Cf. also A349382.

f[p_, e_] := -2*(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A063441(n) = (moebius(n)*sigma(n)); \\ Also Dirichlet inverse of A003959.
A349355(n) = sumdiv(n,d,A003958(n/d)*A063441(d));

a(n) = Sum_{d|n} A003958(n/d) * A063441(d).

Multiplicative with a(p^e) = -2*(p-1)^(e-1). - Amiram Eldar, Nov 16 2021

A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.

Original entry on oeis.org

1, 3, 4, 9, 6, 12, 8, 27, 16, 18, 12, 36, 14, 24, 24, 81, 18, 48, 20, 54, 32, 36, 24, 108, 36, 42, 64, 72, 30, 72, 32, 243, 48, 54, 48, 144, 38, 60, 56, 162, 42, 96, 44, 108, 96, 72, 48, 324, 64, 108, 72, 126, 54, 192, 72, 216, 80, 90, 60, 216, 62, 96, 128, 729, 84, 144, 68

Offset: 1

Marc LeBrun

Completely multiplicative.

Sum of divisors of n with multiplicity. If n = p^m, the number of ways to make p^k as a divisor of n is C(m,k); and sum(C(m,k)*p^k) = (p+1)^k. The rest follows because the function is multiplicative. - Franklin T. Adams-Watters, Jan 25 2010

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Index to divisibility sequences

Apart from initial terms, same as A064478.
Cf. A003958, A027746, A036987, A061142, A163407.
Cf. A063441, A165824, A165825, A165826, A165827.
Cf. A165828, A165829, A165831, A167350, A166590.
Cf. A166642, A166643, A166644, A166645, A166646.
Cf. A166647, A166649, A166650, A166586, A165830.
Cf. A167351, A168065, A168066.

a003959 1 = 1
a003959 n = product $ map (+ 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012

a:= n-> mul((i[1]+1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]]+1)^fi[[All, 2]])); a /@ Range[67] (* Jean-François Alcover, Apr 22 2011 *)

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

Multiplicative with a(p^e) = (p+1)^e. - David W. Wilson, Aug 01 2001
Sum_{n>0} a(n)/n^s = Product_{p prime} 1/(1-p^(-s)-p^(1-s)) (conjectured). - Ralf Stephan, Jul 07 2013
This follows from the absolute convergence of the sum (compare with a(n) = n^2) and the Euler product for completely multiplicative functions. Convergence occurs for at least Re(s)>3. - Thomas Anton, Jul 15 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065488/2 = 1/(2*A005596) = 1.3370563627850107544802059152227440187511993141988459926... - Vaclav Kotesovec, Jul 17 2021
From Thomas Scheuerle, Jul 19 2021: (Start)
a(n) = gcd(A166642(n), A166643(n)).
a(n) = A166642(n)/A061142(n).
a(n) = A166643(n)/A165824(n).
a(n) = A166644(n)/A165825(n).
a(n) = A166645(n)/A165826(n).
a(n) = A166646(n)/A165827(n).
a(n) = A166647(n)/A165828(n).
a(n) = A166649(n)/A165830(n).
a(n) = A166650(n)/A165831(n).
a(n) = A167351(n)/A166590(n). (End)
Dirichlet g.f.: zeta(s-1) * Product_{primes p} (1 + 1/(p^s - p - 1)). - Vaclav Kotesovec, Aug 22 2021

Definition reedited (with formula) by Daniel Forgues, Nov 17 2009

A160889 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.

Original entry on oeis.org

1, 7, 13, 28, 31, 91, 57, 112, 117, 217, 133, 364, 183, 399, 403, 448, 307, 819, 381, 868, 741, 931, 553, 1456, 775, 1281, 1053, 1596, 871, 2821, 993, 1792, 1729, 2149, 1767, 3276, 1407, 2667, 2379, 3472, 1723, 5187, 1893, 3724, 3627, 3871, 2257, 5824, 2793

Offset: 1

N. J. A. Sloane, Nov 19 2009

Dirichlet convolution of A000290 and the series of absolute values of A063441. - R. J. Mathar, Jun 20 2011

a(n) is the number of lattices L in Z^3 such that the quotient group Z^3 / L is C_nm x C_m x C_m (and also C_nm x C_nm x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

			There are 35 = A160870(4,3) lattices of volume 4 in Z^3. Among them, 28 give the quotient group C_4 and 7 give the quotient group C_2 x C_2. Hence, a(4) = 28 and a(2) = 7.
There are 2667 = A160870(32,3) lattices of volume 32 in Z^3. Among them, a(32) = 1792 give the quotient group C_32 (m=1); a(4) = 28 give C_8 x C_2 x C_2 (m=2); a(2) = 7 give C_4 x C_4 x C_2 (m=2).

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index to Jordan function ratios J_k/J_1

Cf. A156304, A000203.

A160889[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(4-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Aug 22 2010 *)

vector(100, n, sumdiv(n^2, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n^2/d), 0))) \\ Altug Alkan, Oct 30 2015

Moebius transform of A064969. Multiplicative with a(p^e) = (p^2+p+1)*p^(2*e-2). - Vladeta Jovovic, Nov 21 2009
a(n) = J_3(n)/J_1(n)=J_3(n)/phi(n)=A059376(n)/A000010(n), where J_k is the k-th Jordan Totient Function. - Enrique Pérez Herrero, Aug 22 2010
Dirichlet g.f.: zeta(s-2)*product_{primes p} (1+p^(1-s)+p^(-s)). - R. J. Mathar, Jun 20 2011
From Álvar Ibeas, Oct 30 2015: (Start)
a(n) = A254981(n^2). For squarefree n, a(n) = A000203(n^2).
a(n) = Sum_{d|n, n/d squarefree} d^2 * A000203(n/d).
(End)
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.748932997843245303033906997685114802259883493595480897273662144... - Vaclav Kotesovec, Dec 18 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/((p^2-1) * (p^2 + p + 1))) = 1.400940662893945919882073637564538872630336562726971915578687405304250550... - Vaclav Kotesovec, Sep 19 2020
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^3). - Ridouane Oudra, Mar 26 2025

Definition corrected by Vladeta Jovovic, Nov 21 2009

Typo in Mathematica program and formula fixed by Enrique Pérez Herrero, Oct 19 2010

A062953 Multiplicative with a(p^e) = -p.

Original entry on oeis.org

1, -2, -3, -2, -5, 6, -7, -2, -3, 10, -11, 6, -13, 14, 15, -2, -17, 6, -19, 10, 21, 22, -23, 6, -5, 26, -3, 14, -29, -30, -31, -2, 33, 34, 35, 6, -37, 38, 39, 10, -41, -42, -43, 22, 15, 46, -47, 6, -7, 10, 51, 26, -53, 6, 55, 14, 57, 58, -59, -30, -61, 62, 21, -2, 65, -66, -67, 34, 69, -70, -71, 6, -73, 74, 15, 38, 77, -78, -79

Offset: 1

Vladeta Jovovic, Jul 21 2001

Except for first term, row products of A142971. - Mats Granvik and Gary W. Adamson, Jul 15 2008

Dirichlet inverse of A003968. - Werner Schulte, Oct 25 2018

Ivan Neretin, Table of n, a(n) for n = 1..10000

Apart from signs, essentially same as A007947.
Cf. A001221, A001615, A003968, A063441, A142971.
Cf. A000010, A023900.

with(numtheory): seq(coeff(series(add(mobius(k)*sigma(k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 80); # Muniru A Asiru, Oct 26 2018

A062953[n_]:=DivisorSum[n,MoebiusMu[ # ]*DivisorSigma[1,# ]&] (* Enrique Pérez Herrero, Aug 24 2010 *)

a(n) = sumdiv(n, d, moebius(d)*sigma(d)); \\ Michel Marcus, Feb 19 2017

a(n) = Sum_{d|n} mu(d)*sigma(d) = Sum_{d|n} A063441(d).
From Enrique Pérez Herrero, Aug 24 2010: (Start)
a(n) = Sum_{d|n} mu(d)*psi(d), where psi is A001615.
a(n) = rad(n)*(-1)^omega(n) = A007947(n)*(-1)^A001221(n). (End)
G.f.: Sum_{k>=1} mu(k)*sigma(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Feb 19 2017
a(n) = (n*invphi(n))/phi(n) = (n*A023900(n))/(A000010(n)). - Amrit Awasthi, Jan 30 2022
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(s-1)). - Amiram Eldar, Sep 18 2023

A349621 Dirichlet convolution of A003415 with the Dirichlet inverse of A003959.

Original entry on oeis.org

0, 1, 1, 1, 1, -2, 1, 0, 2, -2, 1, -3, 1, -2, -2, -4, 1, -5, 1, -3, -2, -2, 1, -4, 4, -2, 3, -3, 1, 3, 1, -16, -2, -2, -2, -7, 1, -2, -2, -4, 1, 3, 1, -3, -5, -2, 1, -4, 6, -9, -2, -3, 1, -12, -2, -4, -2, -2, 1, 5, 1, -2, -5, -48, -2, 3, 1, -3, -2, 3, 1, -8, 1, -2, -9, -3, -2, 3, 1, -4, 0, -2, 1, 5, -2, -2, -2, -4

Offset: 1

Antti Karttunen, Nov 25 2021

sign

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

Cf. A003415, A003959, A063441.

Cf. also A349173, A349394, A349618, A349619, A349620.

f[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, MoebiusMu[#] * DivisorSigma[1, #] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 25 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A063441(n) = (moebius(n)*sigma(n)); \\ Also Dirichlet inverse of A003959.
A349621(n) = sumdiv(n,d,A003415(n/d)*A063441(d));

a(n) = Sum_{d|n} A003415(n/d) * A063441(d).

A128408 Triangle read by rows: A128407 * A051731 as infinite lower triangular matrices.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 01 2007

Left and right borders = mu(n), A008683. Row sums = A008966: (1, -2, -2, 0, -2, 4, -2, 0, 0, 4, ...). A128408 * [1,2,3,...] = A063441: (1, -3, -4, 0, -6, 12, ...). A054524 = A051731 * A128407.

			First few rows of the triangle:
   1;
  -1, -1;
  -1,  0, -1;
   0,  0,  0,  0;
  -1,  0,  0,  0, -1;
   1,  1,  1,  0,  0,  1;
  ...

Cf. A008683, A008966, A054524, A063441, A128407.

a(45) = 0 inserted and more terms from Georg Fischer, Jun 05 2023

A349357 a(n) = A349355(n) + A349356(n).

Original entry on oeis.org

2, 0, 0, 4, 0, 8, 0, 16, 4, 8, 0, 16, 0, 8, 8, 52, 0, 24, 0, 16, 8, 8, 0, 40, 4, 8, 24, 16, 0, 0, 0, 160, 8, 8, 8, 56, 0, 8, 8, 40, 0, 0, 0, 16, 24, 8, 0, 112, 4, 40, 8, 16, 0, 80, 8, 40, 8, 8, 0, 16, 0, 8, 24, 484, 8, 0, 0, 16, 8, 0, 0, 152, 0, 8, 40, 16, 8, 0, 0, 112, 112, 8, 0, 16, 8, 8, 8, 40, 0, 16, 8, 16

Offset: 1

Antti Karttunen, Nov 16 2021

nonn

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

Cf. A003958, A003959, A063441, A097945, A349355, A349356.

f1[p_, e_] := -2*(p - 1)^(e - 1); f2[p_, e_] := 2*(p + 1)^(e - 1); a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A349357(n) = (A349355(n)+A349356(n)); \\ Needs also code from A349355 and A349356.

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349355(d) * A349356(n/d). [As the sequences are Dirichlet inverses of each other]

A343442 If n = Product (p_j^k_j) then a(n) = Product (p_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 4, 5, 4, 7, 20, 9, 4, 5, 28, 13, 20, 15, 36, 35, 4, 19, 20, 21, 28, 45, 52, 25, 20, 7, 60, 5, 36, 31, 140, 33, 4, 65, 76, 63, 20, 39, 84, 75, 28, 43, 180, 45, 52, 35, 100, 49, 20, 9, 28, 95, 60, 55, 20, 91, 36, 105, 124, 61, 140, 63, 132, 45, 4, 105, 260, 69, 76, 125, 252

Offset: 1

Ilya Gutkovskiy, Apr 15 2021

Cf. A000203, A001615, A007429, A007947, A008683, A048250, A062953, A063441, A074816, A092261, A107758, A107759, A319132, A335005.

a[1] = 1; a[n_] := Times @@ ((#[[1]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 70}]
nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k]^2 DivisorSigma[1, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, moebius(d)^2 * sigma(d)) \\ Andrew Howroyd, Apr 15 2021

G.f.: Sum_{k>=1} mu(k)^2 * sigma(k) * x^k / (1 - x^k), where mu = A008683 and sigma = A000203.
a(n) = Sum_{d|n} mu(d)^2 * sigma(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/(12*zeta(3)) = 0.684216... (A335005). - Amiram Eldar, Nov 13 2022
a(n) = Sum_{d|n} mu(d)^2*psi(d), where psi is A001615. - Ridouane Oudra, Jul 24 2025

A143255 Triangle read by rows, A128407 * A126988; 1<=k<=n.

Original entry on oeis.org

1, -2, -1, -3, 0, -1, 0, 0, 0, 0, -5, 0, 0, 0, -1, 6, 3, 2, 0, 0, 1, -7, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 5, 0, 0, 2, 0, 0, 0, 0, 1, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 14, 7, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 02 2008

Left border = A055615: (1, -2, -3, 0, -5, 6, -7,...).
Right border = A008683, mu(n).
Row sums = A063441: (1, -3, -4, 0, -6, 12, 8,...).

			Triangle begins:
  1;
  -2, -1;
  -3, 0, -1;
  0, 0, 0, 0;
  -5, 0, 0, 0, -1;
  6, 3, 2, 0, 0, 1;
  -7, 0, 0, 0, 0, 0, -1;
  0, 0, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 0, 0, 0;
  10, 5, 0, 0, 2, 0, 0, 0, 0, 1;
  ...

Cf. A008683, A063441, A055615, A128407, A126988.

Cf. A128988.

t[n_, m_] = MoebiusMu[n]*If[m == 1, n, If[Mod[n, m] == 0, n/m, 0]]; Table[t[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Roger L. Bagula, Sep 06 2008 *)

Triangle read by rows, A128407 * A126988; 1<=k<=n.

t(n,m) = MoebiusMu(n) * A126988(n,m); t(n,m) = MoebiusMu(n) * if(m == 1, n, if(n mod m == 0, n/m, 0)). - Roger L. Bagula, Sep 06 2008

A062562 a(n) = Sum_{k=1..n} mu(k)*sigma(k).

Original entry on oeis.org

0, 1, -2, -6, -6, -12, 0, -8, -8, -8, 10, -2, -2, -16, 8, 32, 32, 14, 14, -6, -6, 26, 62, 38, 38, 38, 80, 80, 80, 50, -22, -54, -54, -6, 48, 96, 96, 58, 118, 174, 174, 132, 36, -8, -8, -8, 64, 16, 16, 16, 16, 88, 88, 34, 34, 106, 106, 186, 276, 216, 216, 154, 250, 250, 250, 334, 190, 122, 122, 218, 74, 2, 2, -72, 42, 42, 42

Offset: 0

Jason Earls, Jul 03 2001

sign

Partial sums of A063441.

Table[Sum[MoebiusMu[k]DivisorSigma[1,k],{k,n}],{n,0,80}] (* Harvey P. Dale, Nov 08 2011 *)

s=[]; for(n=0,130,s=concat(s,sum(k=1,n,moebius(k)*sigma(k)))); s

Corrected by Harvey P. Dale, Nov 08 2011

cp's OEIS Frontend

A349355 Dirichlet convolution of A003958 with A063441 (Dirichlet inverse of A003959), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

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A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.

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A160889 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.

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A062953 Multiplicative with a(p^e) = -p.

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A349621 Dirichlet convolution of A003415 with the Dirichlet inverse of A003959.

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A128408 Triangle read by rows: A128407 * A051731 as infinite lower triangular matrices.

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A349357 a(n) = A349355(n) + A349356(n).

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