cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A018806 Sum of gcd(x, y) for 1 <= x, y <= n.

Original entry on oeis.org

1, 5, 12, 24, 37, 61, 80, 112, 145, 189, 220, 288, 325, 389, 464, 544, 593, 701, 756, 880, 989, 1093, 1160, 1336, 1441, 1565, 1700, 1880, 1965, 2205, 2296, 2488, 2665, 2829, 3028, 3328, 3437, 3621, 3832, 4152, 4273, 4621, 4748, 5040, 5373, 5597, 5736, 6168
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

a(n) is also the entrywise 1-norm of the n X n GCD matrix.

Links

Crossrefs

Cf. A000010, A018805, A064951, A344479.

Programs

  • Maple
    N:= 1000 # to get a(1) to a(N)
g:= add(numtheory:-phi(k)*x^k*(1+x^k)/((1-x^k)^2*(1-x)),k=1..N):
S:= series(g, x, N+1):
seq(coeff(S,x,j), j=1..N); # Robert Israel, Jan 14 2015
  • Mathematica
    Table[nn = n;Total[Level[Table[Table[GCD[i, j], {i, 1, nn}], {j, 1, nn}], {2}]], {n, 1, 48}] (* Geoffrey Critzer, Jan 14 2015 *)
  • PARI
    a(n)=2*sum(i=1,n,sum(j=1,i-1,gcd(i,j)))+n*(n+1)/2 \\ Charles R Greathouse IV, Jun 21 2013
  • PARI
    a(n)=sum(k=1,n,eulerphi(k)*(n\k)^2) \\ Charles R Greathouse IV, Jun 21 2013
  • Python
    from sympy import totient
def A018806(n): return sum(totient(k)*(n//k)**2 for k in range(1,n+1)) # Chai Wah Wu, Aug 05 2024

Formula

Sum_{k=1..n} phi(k)*(floor(n/k))^2. - Vladeta Jovovic, Nov 10 2002
a(n) ~ kn^2 log n, with k = 6/Pi^2. - Charles R Greathouse IV, Jun 21 2013
G.f.: Sum_{k >= 1} phi(k)*x^k*(1+x^k)/((1-x^k)^2*(1-x)). - Robert Israel, Jan 14 2015

A344522 a(n) = Sum_{1 <= i, j, k <= n} gcd(i,j,k).

Original entry on oeis.org

1, 9, 30, 76, 141, 267, 400, 624, 885, 1249, 1590, 2208, 2689, 3411, 4248, 5248, 6081, 7485, 8530, 10248, 11889, 13687, 15228, 17988, 20053, 22569, 25242, 28588, 31053, 35463, 38284, 42540, 46581, 50893, 55362, 61824, 65857, 71247, 76884, 84388, 89349, 97881, 103342
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Column k=3 of A344479.
Cf. A018806, A071778, A343497, A344132, A344521, A344523, A344524, A344525, A344526.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, k]))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^3);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^3.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.
a(n) ~ Pi^2 * n^3 / (6*zeta(3)). - Vaclav Kotesovec, May 23 2021

A344527 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of ordered k-tuples (x_1, x_2, ..., x_k) with gcd(x_1, x_2, ..., x_k) = 1 (1 <= {x_1, x_2, ..., x_k} <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 11, 1, 1, 31, 79, 55, 19, 1, 1, 63, 241, 239, 115, 23, 1, 1, 127, 727, 991, 607, 181, 35, 1, 1, 255, 2185, 4031, 3091, 1199, 307, 43, 1, 1, 511, 6559, 16255, 15559, 7501, 2303, 439, 55, 1, 1, 1023, 19681, 65279, 77995, 45863, 16531, 3823, 637, 63, 1
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Examples

			G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} mu(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  3,   7,   15,   31,    63, ...
  1,  7,  25,   79,  241,   727, ...
  1, 11,  55,  239,  991,  4031, ...
  1, 19, 115,  607, 3091, 15559, ...
  1, 23, 181, 1199, 7501, 45863, ...

Links

Crossrefs

Columns k=1..6 give A000012, A018805, A071778, A082540, A082544, A343978.
T(n,n) gives A332468.
Cf. A343510, A343516, A344479.

Programs

  • Mathematica
    T[n_, k_] := Sum[MoebiusMu[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, moebius(j)*(n\j)^k);
  • PARI
    T(n, k) = n^k-sum(j=2, n, T(n\j, k));
  • Python
    from functools import lru_cache
from itertools import count, islice
@lru_cache(maxsize=None)
def A344527_T(n,k):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A344527_T(k1,k)
        j, k1 = j2, n//j2
    return n*(n**(k-1)-1)-c+j
def A344527_gen(): # generator of terms
    return (A344527_T(k+1, n-k) for n in count(1) for k in range(n))
A344527_list = list(islice(A344527_gen(),30)) # Chai Wah Wu, Nov 02 2023

Formula

G.f. of column k: (1/(1 - x)) * Sum_{i>=1} mu(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
T(n,k) = Sum_{j=1..n} mu(j) * floor(n/j)^k.
T(n,k) = n^k - Sum_{j=2..n} T(floor(n/j),k).

A344523 a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i,j,k,l).

Original entry on oeis.org

1, 17, 84, 276, 649, 1417, 2528, 4432, 7033, 10905, 15556, 22836, 30673, 41729, 54944, 71968, 89969, 115457, 140820, 175444, 212537, 257113, 302720, 366160, 426505, 500873, 580676, 677108, 769761, 895377, 1008928, 1153120, 1300417, 1469073, 1640020, 1860340, 2054921
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Column k=4 of A344479.
Cf. A082540, A343498, A344138, A344522, A344524, A344525.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, gcd([i, j, k, l])))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^4);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^4.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.
a(n) ~ 90 * zeta(3) * n^4 / Pi^4. - Vaclav Kotesovec, May 23 2021

A344524 a(n) = Sum_{1 <= i, j, k, l, m <= n} gcd(i,j,k,l,m).

Original entry on oeis.org

1, 33, 246, 1060, 3165, 8091, 17128, 33936, 60645, 103825, 164886, 259368, 381841, 557595, 784200, 1091056, 1462353, 1968261, 2554810, 3327120, 4230561, 5361463, 6644196, 8302020, 10113445, 12352041, 14873418, 17924356, 21225165, 25341375, 29670556, 34920348, 40625541, 47297365
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Comments

In general, for m > 2, Sum_{k=1..n} phi(k) * floor(n/k)^m ~ zeta(m-1) * n^m / zeta(m). - Vaclav Kotesovec, May 23 2021

Links

Crossrefs

Column k=5 of A344479.
Cf. A082544, A343499, A344139, A344522, A344523, A344525.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, sum(m=1, n, gcd([i, j, k, l, m]))))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^5);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^5)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^5.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^5.
a(n) ~ Pi^4 * n^5 / (90*zeta(5)). - Vaclav Kotesovec, May 23 2021

A344525 a(n) = Sum_{1 <= x_1, x_2, ... , x_n <= n} gcd(x_1,x_2, ... ,x_n).

Original entry on oeis.org

1, 5, 30, 276, 3165, 47521, 826000, 16843792, 387723045, 10009889889, 285360865350, 8918311872516, 302888304741841, 11112685595264369, 437898699063881208, 18447025862624951488, 827242515246907227633, 39346558373191515582161
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Main diagonal of A344479.
Cf. A018806, A344140, A344522, A344523, A344524.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^n, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^n);
  • Python
    from sympy import totient
def A344525(n): return sum(totient(k)*(n//k)**n for k in range(1,n+1)) # Chai Wah Wu, Aug 05 2024

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^n.
a(n) ~ n^n. - Vaclav Kotesovec, May 23 2021

A345229 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1 <= x_2 <= ... <= x_k <= n} gcd(x_1, x_2, ..., x_k).

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 9, 10, 1, 6, 13, 17, 15, 1, 7, 18, 28, 26, 21, 1, 8, 24, 44, 47, 41, 28, 1, 9, 31, 66, 83, 82, 54, 36, 1, 10, 39, 95, 140, 159, 116, 74, 45, 1, 11, 48, 132, 225, 293, 249, 172, 95, 55, 1, 12, 58, 178, 346, 512, 509, 401, 235, 122, 66, 1, 13, 69, 234, 512, 852, 980, 888, 592, 321, 143, 78
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2021

Keywords

Examples

			G.f. of column 3: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^3.
Square array begins:
   1,  1,   1,   1,   1,   1,    1, ...
   3,  4,   5,   6,   7,   8,    9, ...
   6,  9,  13,  18,  24,  31,   39, ...
  10, 17,  28,  44,  66,  95,  132, ...
  15, 26,  47,  83, 140, 225,  346, ...
  21, 41,  82, 159, 293, 512,  852, ...
  28, 54, 116, 249, 509, 980, 1782, ...

Links

Crossrefs

Columns k=1..4 give A000217, A272718, A344521, A344992.
Main diagonal gives A345230.
Cf. A343516, A344479.

Programs

  • Maple
    T:= (n, k)-> coeff(series((1/(1-x))* add(numtheory[phi](j)
             *x^j/(1-x^j)^k, j=1..n), x, n+1), x, n):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jun 11 2021
  • Mathematica
    T[n_, k_] := Sum[DivisorSum[j, EulerPhi[j/#] * Binomial[k + # - 2, k - 1] &], {j, 1, n}];  Table[T[k, n - k + 1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 11 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, sumdiv(j, d, eulerphi(j/d)*binomial(d+k-2, k-1)));
  • PARI
    T(n, k) = sum(j=1, n, eulerphi(j)*binomial(n\j+k-1, k)); \\ Seiichi Manyama, Sep 13 2024

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^k.
T(n,k) = Sum_{j=1..n} Sum_{d|j} phi(j/d) * binomial(d+k-2, k-1).
T(n,k) = Sum_{j=1..n} phi(j) * binomial(floor(n/j)+k-1,k). - Seiichi Manyama, Sep 13 2024
Showing 1-7 of 7 results.

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