A209295 -id:A209295 - cp's OEIS frontend

A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).

1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 4, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 4, 4, 0, 0, 1, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 6

Offset: 1

N. J. A. Sloane, Apr 09 2000

From Gary W. Adamson, Jan 08 2007: (Start)
Let H be this lower triangular matrix. Then:
H * A051731 = A126988,
H * [1, 2, 3, ...] = 1, 3, 5, 8, 9, 15, ... = A018804,
H * sigma(n) = A038040 = d(n) * n = 1, 4, 6, 12, 10, ... where sigma(n) = A000203,
H * d(n) (A000005) = sigma(n) = A000203,
Row sums are A000027 (corrected by Werner Schulte, Sep 06 2020, see comment of Gary W. Adamson, Aug 03 2008),
H^2 * d(n) = d(n)*n, H^2 = A127192,
H * mu(n) (A008683) = A007431(n) (corrected by Werner Schulte, Sep 06 2020),
H^2 row sums = A018804. (End)
The Möbius inversion principle of Richard Dedekind and Joseph Liouville (1857), cf. "Concrete Mathematics", p. 136, is equivalent to the statement that row sums are the row index n. - Gary W. Adamson, Aug 03 2008
The multivariable row polynomials give n times the cycle index for the cyclic group C_n, called Z(C_n) (see the MathWorld link with the Harary reference): n*Z(C_n) = Sum_{k=1..n} T(n,k)*(y_{n/k})^k, n >= 1. E.g., 6*Z(C_6) = 2*(y_6)^1 + 2*(y_3)^2 + 1*(y_2)^3 + 1*(y_1)^6. - Wolfdieter Lang, May 22 2012
See A102190 (no 0's, rows reversed). - Wolfdieter Lang, May 29 2012
This is the number of permutations in the n-th cyclic group which are the product of k disjoint cycles. - Robert A. Beeler, Aug 09 2013

			Triangle begins
   1;
   1, 1;
   2, 0, 1;
   2, 1, 0, 1;
   4, 0, 0, 0, 1;
   2, 2, 1, 0, 0, 1;
   6, 0, 0, 0, 0, 0, 1;
   4, 2, 0, 1, 0, 0, 0, 1;
   6, 0, 2, 0, 0, 0, 0, 0, 1;
   4, 4, 0, 0, 1, 0, 0, 0, 0, 1;
  10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1;

Ronald L. Graham, D. E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, p. 136.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Cycle Index.

Cf. A000005, A000007, A000010, A000012, A000203, A007431, A008683.
Cf. A038040, A051731, A054521, A054525, A102190, A126988.
Sums incliude: A029935, A069097, A092843 (diagonal), A209295.
Sums of the form Sum_{k} k^p * T(n, k): A000027 (p=0), A018804 (p=1), A069097 (p=2), A343497 (p=3), A343498 (p=4), A343499 (p=5).

a054523 n k = a054523_tabl !! (n-1) !! (k-1)
a054523_row n = a054523_tabl !! (n-1)
a054523_tabl = map (map (\x -> if x == 0 then 0 else a000010 x)) a126988_tabl
-- Reinhard Zumkeller, Jan 20 2014

A054523:= func< n,k | k eq n select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
[A054523(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024

A054523 := proc(n,k) if n mod k = 0 then numtheory[phi](n/k) ; else 0; end if; end proc: # R. J. Mathar, Apr 11 2011

T[n_, k_]:= If[k==n,1,If[Divisible[n, k], EulerPhi[n/k], 0]];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Dec 15 2017 *)

for(n=1, 10, for(k=1, n, print1(if(!(n % k), eulerphi(n/k), 0), ", "))) \\ G. C. Greubel, Dec 15 2017

def A054523(n,k):
    if (k==n): return 1
    elif (n%k)==0: return euler_phi(int(n//k))
    else: return 0
flatten([[A054523(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 24 2024

Sum_{k=1..n} k * T(n, k) = A018804(n). - Gary W. Adamson, Jan 08 2007
Equals A054525 * A126988 as infinite lower triangular matrices. - Gary W. Adamson, Aug 03 2008
From Werner Schulte, Sep 06 2020: (Start)
Sum_{k=1..n} T(n,k) * A000010(k) = A029935(n) for n > 0.
Sum_{k=1..n} k^2 * T(n,k) = A069097(n) for n > 0. (End)
From G. C. Greubel, Jun 24 2024: (Start)
T(2*n-1, n) = A000007(n-1), n >= 1.
T(2*n, n) = A000012(n), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1 - (-1)^n)*n/2.
Sum_{k=1..floor(n+1)/2} T(n-k+1, k) = A092843(n+1).
Sum_{k=1..n} (k+1)*T(n, k) = A209295(n).
Sum_{k=1..n} k^3 * T(n, k) = A343497(n).
Sum_{k=1..n} k^4 * T(n, k) = A343498(n).
Sum_{k=1..n} k^5 * T(n, k) = A343499(n). (End)

A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k).

Original entry on oeis.org

0, 0, 1, 4, 8, 20, 21, 56, 60, 96, 105, 220, 152, 364, 301, 360, 464, 816, 549, 1140, 760, 1036, 1221, 2024, 1196, 2200, 2041, 2484, 2184, 4060, 2205, 4960, 3664, 4224, 4641, 5180, 4008, 8436, 6517, 7072, 5980, 11480, 6321, 13244, 8888, 9540, 11661, 17296

Offset: 0

N. J. A. Sloane

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from Reinhard Zumkeller)
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.

Antidiagonal sums of array A003990.
Cf. A209295.
Cf. A000010, A023900, A057660, A130054.

a006580 n = a006580_list !! (n-1)
a006580_list = map sum a003990_tabl
-- Reinhard Zumkeller, Aug 05 2012

a:= n-> add(ilcm(j, n-j), j=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 25 2019

Table[ Sum[ LCM[ k, n-k ], {k, 1, n-1} ], {n, 2, 50} ] (* Olivier Gérard, Aug 15 1997 *)
f1[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); f2[p_, e_] := 1 - (p - 1)*e; a[n_] := (Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct)*n/6; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sum(k=1, n-1, lcm(k, n-k)); \\ Michel Marcus, Aug 11 2017

For n > 0, a(n) = (n/6)*Sum_{d|n} (d*phi(d) - A023900(d)). - Sebastian Karlsson, Oct 02 2021

a(n) = (n/6) * (A057660(n) - A130054(n)), for n > 0. - Amiram Eldar, Apr 28 2023

More terms from Olivier Gérard, Aug 15 1997

A344510 a(n) = Sum_{k=1..n} k * gcd(k,n).

Original entry on oeis.org

1, 5, 12, 24, 35, 63, 70, 112, 135, 185, 176, 312, 247, 371, 450, 512, 425, 729, 532, 920, 903, 935, 782, 1488, 1125, 1313, 1458, 1848, 1247, 2475, 1426, 2304, 2277, 2261, 2660, 3672, 2035, 2831, 3198, 4400, 2501, 4977, 2752, 4664, 5265, 4163, 3290, 6912, 4459, 6125, 5508, 6552

Offset: 1

Seiichi Manyama, May 21 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000217, A018804, A209295, A344508.

A344510:= func< n | (n/2)*(&+[(d+1)*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A344510(n): n in [1..60]]; // G. C. Greubel, Jun 24 2024

a[n_] := Sum[k * GCD[k, n], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 21 2021 *)
A344510[n_]:= (n/2)*DivisorSum[n, (#+1)*EulerPhi[n/#] &];
Table[A344510[n], {n,60}] (* G. C. Greubel, Jun 24 2024 *)

PARI
```
a(n) = sum(k=1, n, k*gcd(k, n));
```

a(n) = n*sumdiv(n, d, eulerphi(n/d)*(d+1))/2;

def A344510(n): return (n/2)*sum((k+1)*euler_phi(int(n//k)) for k in (1..n) if (k).divides(n))
[A344510(n) for n in range(1,61)] # G. C. Greubel, Jun 24 2024

a(n) = n * (n + A018804(n))/2.
a(n) = (n/2) * (n + Sum_{d|n} phi(n/d) * d).
a(n) = (n/2) * Sum_{d|n} phi(n/d) * (d+1).

A327031 A(n, k) = Sum_{d|n} phi(d)*T(n/d, k) if n > 0 and A(0, k) = 0 where T(n, k) = binomial(n+k-1, n). Square array read by ascending antidiagonals, with n, k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 8, 9, 4, 0, 0, 5, 12, 16, 14, 5, 0, 0, 6, 14, 27, 28, 20, 6, 0, 0, 7, 21, 33, 53, 45, 27, 7, 0, 0, 8, 20, 56, 72, 95, 68, 35, 8, 0, 0, 9, 28, 54, 132, 146, 159, 98, 44, 9, 0, 0, 10, 30, 84, 144, 285, 276, 252, 136, 54, 10, 0

Offset: 0

Peter Luschny, Aug 25 2019

			[0] 0, 0,  0,  0,   0,   0,    0,    0,     0,     0, ... A000004
[1] 0, 1,  2,  3,   4,   5,    6,    7,     8,     9, ... A001477
[2] 0, 2,  5,  9,  14,  20,   27,   35,    44,    54, ... A000096
[3] 0, 3,  8, 16,  28,  45,   68,   98,   136,   183, ... A255993 (conj.)
[4] 0, 4, 12, 27,  53,  95,  159,  252,   382,   558, ... A327032
[5] 0, 5, 14, 33,  72, 146,  276,  490,   824,  1323, ...
[6] 0, 6, 21, 56, 132, 285,  572, 1078,  1924,  3276, ...
[7] 0, 7, 20, 54, 144, 360,  828, 1758,  3480,  6489, ...
[8] 0, 8, 28, 84, 236, 615, 1479, 3297,  6869, 13491, ...
[9] 0, 9, 30, 93, 284, 815, 2150, 5215, 11728, 24694, ...
     A209295,

Cf. A000004 (n=0), A001477 (n=1), A000096 (n=2), A255993 (n=3 conj.), A327032 (n=4), A209295, A097805.

DivisorSquareArray := proc(p, T, len) local row:
row := (n, k) -> add(p(d)*T(n/d, k), d = numtheory:-divisors(n)):
seq(lprint(seq(add(j, j=row(n, k)), k=0..len-1)), n=0..len-1) end:
DivisorSquareArray(numtheory:-phi, (n, k) -> binomial(n+k-1, n), 9);

def DivisorSquareArray(p, T, Len):
    D = [[0]*Len]
    for n in (1..Len-1):
        r = lambda k: [p(d)*T(n//d, k) for d in divisors(n)]
        L = [sum(r(k)) for k in (0..Len-1)]
        D.append(L)
    return D
def T(n, k): return binomial(n + k - 1, n)
DivisorSquareArray(euler_phi, T, 10)

cp's OEIS Frontend

A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).

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A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k).

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A344510 a(n) = Sum_{k=1..n} k * gcd(k,n).

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A327031 A(n, k) = Sum_{d|n} phi(d)*T(n/d, k) if n > 0 and A(0, k) = 0 where T(n, k) = binomial(n+k-1, n). Square array read by ascending antidiagonals, with n, k >= 0.

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