A018804 -id:A018804 - cp's OEIS frontend

A209295 Antidiagonal sums of the gcd(.,.) array A109004.

0, 2, 5, 8, 12, 14, 21, 20, 28, 30, 37, 32, 52, 38, 53, 60, 64, 50, 81, 56, 92, 86, 85, 68, 124, 90, 101, 108, 132, 86, 165, 92, 144, 138, 133, 152, 204, 110, 149, 164, 220, 122, 237, 128, 212, 234, 181, 140, 288, 182, 245, 216, 252, 158, 297, 244

Offset: 0

R. J. Mathar, Jan 17 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5000 terms from G. C. Greubel)

Cf. A006580, A018804, A109004.

A209295:= func< n | n eq 0 select 0 else (&+[(n/d+1)*EulerPhi(d): d in Divisors(n)]) >;
[A209295(n): n in [0..40]]; // G. C. Greubel, Jun 24 2024

a:= n-> add(igcd(j, n-j), j=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 25 2019
# Alternative (computes [a(n), n=0..10000] about 25 times faster):
a := n -> add(numtheory:-phi(d)*(n/d + 1), d = numtheory:-divisors(n)):
seq(a(n), n = 0..57); # Peter Luschny, Aug 25 2019

Table[Sum[GCD[n-k,k], {k,0,n}], {n,0,50}] (* G. C. Greubel, Jan 04 2018 *)
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := n + Times @@ f @@@ FactorInteger[n]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Apr 28 2023 *)

a(n) = n + sum(k=1, n, gcd(n,k)); \\ Michel Marcus, Jan 05 2018

def A209295(n): return sum((n/k+1)*euler_phi(k) for k in (1..n) if (k).divides(n))
[A209295(n) for n in range(41)] # G. C. Greubel, Jun 24 2024

a(0) = 0; a(n) = A018804(n) + n for n > 0. [Amended by Georg Fischer, Jan 25 2020]

a(n) = Sum_{d|n} phi(d)*(n/d + 1) for n >= 1. - Peter Luschny, Aug 25 2019

A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 2, 6, 14, 40, 84, 224, 484, 1134, 2480, 5632, 12036, 26624, 56896, 122640, 261078, 557056, 1176876, 2490368, 5237360, 11008704, 23057408, 48234496, 100635144, 209714400, 436154368, 905962860, 1878931264, 3892314112, 8052800160, 16642998272, 34359209436

Offset: 1

Gus Wiseman, Feb 03 2018

nonn

For prime p, a(p) = 2^(p-2)*p. - Jon E. Schoenfield, Feb 03 2018

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A001787, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299149, A299151.

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1, n*2^(n-2)-
       add(a(d)*a(n/d), d=divisors(n) minus {1, n})/2)
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Mar 07 2018

nn=50;
sys=Table[2^(n-1)*n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]

A309322 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 4, 8, 15, 19, 35, 34, 56, 63, 86, 76, 141, 103, 157, 182, 212, 169, 294, 208, 355, 335, 359, 298, 556, 405, 490, 522, 657, 463, 865, 526, 816, 773, 812, 856, 1239, 739, 1003, 1058, 1424, 901, 1610, 988, 1525, 1617, 1445, 1174, 2188, 1435, 1960, 1760, 2091, 1483, 2529, 1994

Offset: 1

Ilya Gutkovskiy, Jul 23 2019

nonn

Dirichlet convolution of Euler totient function with triangular numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000010, A000217, A018804, A069097, A272718, A309323.

nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[EulerPhi[n/d] d (d + 1)/2, {d, Divisors[n]}], {n, 1, 55}]
Table[Sum[Sum[GCD[j, k, n], {j, 1, k}], {k, 1, n}], {n, 1, 55}]

a(n) = Sum_{d|n} phi(n/d) * d * (d + 1)/2.
a(n) = Sum_{k=1..n} Sum_{j=1..k} gcd(j,k,n).
a(n) = Sum_{k=1..n} gcd(n,k)*(gcd(n,k)+1)/2. - Richard L. Ollerton, May 07 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (36*zeta(3)). - Vaclav Kotesovec, May 23 2021
a(n) = (A018804(n) + A069097(n))/2. - Ridouane Oudra, May 22 2025

A309323 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 5, 12, 26, 39, 76, 90, 152, 191, 275, 296, 492, 467, 674, 798, 1000, 985, 1467, 1348, 1934, 2011, 2360, 2322, 3420, 3085, 3791, 4062, 4944, 4523, 6454, 5486, 7168, 7237, 8189, 8340, 10942, 9175, 11300, 11714, 14208, 12381, 16759, 14232, 18036, 18549, 19706, 18470

Offset: 1

Ilya Gutkovskiy, Jul 23 2019

nonn

Dirichlet convolution of Euler totient function with tetrahedral numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000010, A000292, A018804, A272718, A309322.

nmax = 47; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[EulerPhi[n/d] d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 1, 47}]
Table[Sum[Sum[Sum[GCD[i, j, k, n], {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 1, 47}]

a(n) = Sum_{d|n} phi(n/d) * d * (d + 1) * (d + 2)/6.
a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} gcd(i,j,k,n).
Sum_{k=1..n} a(k) ~ 15 * zeta(3) * n^4 / (4*Pi^4). - Vaclav Kotesovec, May 23 2021

A081000 n is a member if and only if it ranks among top n positive integers in centrality (cf. A080997 for fuller description of this concept).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120

Offset: 1

Matthew Vandermast, Mar 02 2003

nonn

1, 2, 3, 4, 21 and 27 are currently the only known examples of n that rank exactly n-th in centrality; it is not known whether there are others.

Cf. A080997, A080998, also A081029, the highly central integers (a subset of this sequence). Complement is A081001.

The formula for the centrality of an integer is A018804(n)/n^2; see also A080997.

A127172 Cube of A051731.

Original entry on oeis.org

1, 3, 1, 3, 0, 1, 6, 3, 0, 1, 3, 0, 0, 0, 1, 9, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 10, 6, 0, 3, 0, 0, 0, 1, 6, 0, 3, 0, 0, 0, 0, 0, 1, 9, 3, 0, 0, 0, 3, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Nonzero terms in every column = A007425: (1, 3, 3, 6, 3, 9, 3, ...).
Row sums = A007426: (1, 4, 4, 20, 4, 16, ...).
A127172 * mu(n) = d(n); or A127172 * A008683 = A000005.
A127172 * d(n) = tau_5(n); or A127172 * A000005 = A061200.
A127172 * phi(n) = A007429: (1, 4, 5, 11, 7, 20, ...); or: A127172 * A000010 = A007429.
Note that A051731 * d(n) = row sums of A127172; or A051731 * A000005 = A007425.
Also, A126988 * mu(n) = phi(n); or A126988 * A008683 = A000010.
A126988 * phi(n) = A018804: (1, 3, 5, 8, 9, 15, ...); = A127170 * mu(n).

			First few rows of the triangle:
   1;
   3, 1;
   3, 0, 1;
   6, 3, 0, 1;
   3, 0, 0, 0, 1;
   9, 3, 3, 0, 0, 1;
   3, 0, 0, 0, 0, 0, 1;
  10, 6, 0, 3, 0, 0, 0, 1;
   6, 0, 3, 0, 0, 0, 0, 0, 1;
   9, 3, 0, 0, 3, 0, 0, 0, 0, 1;
  ...

Cf. A000005, A007425, A127170, A051731, A007429, A000010, A126988, A018804.

Cube of A051731 A007425: (1, 3, 3, 6, 3, 9, 3, ...) in every column k, interspersed with (k-1) zeros.

A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

Original entry on oeis.org

1, 2, 1, 3, 0, 2, 4, 2, 0, 2, 5, 0, 0, 0, 4, 6, 3, 4, 0, 0, 2, 7, 0, 0, 0, 0, 0, 6, 8, 4, 0, 4, 0, 0, 0, 4, 9, 0, 6, 0, 0, 0, 0, 0, 6, 10, 5, 0, 0, 8, 0, 0, 0, 0, 4, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 12, 6, 8, 6, 0, 4, 0, 0, 0, 0, 0, 4, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 14, 7, 0, 0, 0, 0, 12, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

If the two matrices A054523 and A054522 are commuted, the matrix product becomes A127477.

			First few rows of the triangle are:
.1;
.2, 1;
.3, 0, 2;
.4, 2, 0, 2;
.5, 0, 0, 0, 4;
.6, 3, 4, 0, 0, 2;
.7, 0, 0, 0, 0, 0, 6;
.8, 4, 0, 4, 0, 0, 0, 4;
....

Cf. A054522, A054523, A018804, A000010.

A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127478 := proc(n,k) add( A054523(n,j)*A054522(j,k), j=k..n) ; end: seq(seq( A127478(n,k),k=1..n),n=1..15) ;

T(n,k) = sum_{j=k..n} A054523(n,j) * A054522(j,k).
T(n,n) = A000010(n) (diagonal).
sum_{k=1..n} T(n,k) = A018804(n) (row sums).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

A294402 E.g.f.: exp(-Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, -1, -3, -1, 1, 279, 301, 12263, 5601, -431281, -2140739, -77720721, -1755429983, -12569445721, 85768062381, -4458503862121, 43351731658561, 546719071653663, 31735514726673661, 291860504886837599, 5860390638855992001, 208620917963122666679

Offset: 0

Seiichi Manyama, Oct 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..448

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294403 (k=1), A294404 (k=2).

Cf. A000005, A018804, A028343, A294363.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j). - Ilya Gutkovskiy, Aug 17 2021

A294459 E.g.f.: exp(-Sum_{n>=1} A001227(n) * x^n).

Original entry on oeis.org

1, -1, -1, -7, 25, -41, 631, 881, 98897, -609265, 3798991, -41799671, 914146729, -15008576857, 16469525255, -5181463756351, 79515495724321, -1220435382764129, 12608713897126687, -449855614172366695, 10437031873016276921, -231918657853281955081

Offset: 0

Seiichi Manyama, Oct 31 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..446

E.g.f.: exp(-Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): this sequence (k=0), A294460 (k=1), A294461 (k=2).

Cf. A018804.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d%2)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A001227(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} 1 / (1 + x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j). - Ilya Gutkovskiy, Aug 17 2021

A332794 a(n) = Sum_{d|n} (-1)^(d + 1) * d * phi(n/d).

Original entry on oeis.org

1, -1, 5, -4, 9, -5, 13, -12, 21, -9, 21, -20, 25, -13, 45, -32, 33, -21, 37, -36, 65, -21, 45, -60, 65, -25, 81, -52, 57, -45, 61, -80, 105, -33, 117, -84, 73, -37, 125, -108, 81, -65, 85, -84, 189, -45, 93, -160, 133, -65, 165, -100, 105, -81, 189

Offset: 1

Ilya Gutkovskiy, Feb 24 2020

Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.

Cf. A000010, A018804, A078747, A143520, A181983, A193356, A199084, A344372, A368929.

a[n_] := Sum[(-1)^(d + 1) d EulerPhi[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := If[OddQ[n], Sum[GCD[n, k], {k, 1, n}], Sum[(-1)^(k + 1) GCD[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e*(p-1) + p)*p^(e-1); f[2, e_] := -e*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2022 *)

a(n) = sumdiv(n, d, (-1)^(d+1)*d*eulerphi(n/d)); \\ Michel Marcus, Feb 24 2020

G.f.: Sum_{k>=1} phi(k) * x^k / (1 + x^k)^2.
Dirichlet g.f.: zeta(s-1)^2 * (1 - 2^(2 - s)) / zeta(s).
a(n) = Sum_{k=1..n} gcd(n, k) if n odd, Sum_{k=1..n} (-1)^(k + 1) * gcd(n, k) if n even.
From Amiram Eldar, Nov 04 2022: (Start)
Multiplicative with a(2^e) = -e*2^(e-1), and a(p^e) = (e*(p-1) + p)*p^(e-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*log(2)/Pi^2 = 0.210691... . (End)
a(2*n) = - Sum_{k = 1..n} gcd(2*k, n) = - A344372(n); a(2*n+1) = A018804(2*n+1). - Peter Bala, Jan 11 2024
a(n) = Sum_{k = 1..n} (-1)^(1 + gcd(k, n)) * gcd(k, n) (follows from an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

cp's OEIS Frontend

A209295 Antidiagonal sums of the gcd(.,.) array A109004.

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A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

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A309322 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi = Euler totient function (A000010).

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A309323 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi = Euler totient function (A000010).

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A081000 n is a member if and only if it ranks among top n positive integers in centrality (cf. A080997 for fuller description of this concept).

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A127172 Cube of A051731.

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A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

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A294402 E.g.f.: exp(-Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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A294459 E.g.f.: exp(-Sum_{n>=1} A001227(n) * x^n).