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 1-6 of 6 results.

A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 10, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 27, 24, 28, 8, 15, 24, 30, 42, 42, 30, 24, 15, 13, 31, 32, 60, 16, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 28, 51, 48, 70, 48, 51, 28, 42, 12, 28, 36
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).
T(n,1) = T(1,n) = A000203(n) = sigma(n).
T(n,2) = T(2,n) = A062731(n) = sigma(2*n).
T(n+1,n) = A083539(n) = sigma(n+1)*sigma(n).
T(prime(n),1) = A008864(n) = prime(n)+1.

Examples

			[-----1---2---3----4----5----6----7----8----9---10---11---12]
[ 1]  1,  3,  4,   7,   6,  12,   8,  15,  13,  18,  12,  28
[ 2]  3,  7, 12,  15,  18,  28,  24,  31,  39,  42,  36,  60
[ 3]  4, 12, 10,  28,  24,  30,  32,  60,  28,  72,  48,  70
[ 4]  7, 15, 28,  27,  42,  60,  56,  51,  91,  90,  84, 108
[ 5]  6, 18, 24,  42,  16,  72,  48,  90,  78,  48,  72, 168
[ 6] 12, 28, 30,  60,  72,  70,  96, 124,  84, 168, 144, 150
[ 7]  8, 24, 32,  56,  48,  96,  22, 120, 104, 144,  96, 224
[ 8] 15, 31, 60,  51,  90, 124, 120,  83, 195, 186, 180, 204
[ 9] 13, 39, 28,  91,  78,  84, 104, 195,  55, 234, 156, 196
[10] 18, 42, 72,  90,  48, 168, 144, 186, 234, 112, 216, 360
[11] 12, 36, 48,  84,  72, 144,  96, 180, 156, 216,  34, 336
[12] 28, 60, 70, 108, 168, 150, 224, 204, 196, 360, 336, 270
.
Displayed as a triangular array:
    1;
    3,  3;
    4,  7,  4;
    7, 12, 12,  7;
    6, 15, 10, 15,  6;
   12, 18, 28, 28, 18, 12;
    8, 28, 24, 27, 24, 28,  8;
   15, 24, 30, 42, 42, 30, 24, 15;
   13, 31, 32, 60, 16, 60, 32, 31, 13;

Links

Crossrefs

Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216627.

Programs

  • Maple
    with(numtheory):
T:= (n, k) -> add(add(ilcm(c, d), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    T[n_, k_] := Sum[LCM[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)
  • Sage
    def A216626(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(lcm, cp))
for n in (1..12): [A216626(n,k) for k in (1..12)]

A216627 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = sum_{c|n,d|k} lcm(c,d).

Original entry on oeis.org

1, 3, 7, 4, 12, 10, 7, 15, 28, 27, 6, 18, 24, 42, 16, 12, 28, 30, 60, 72, 70, 8, 24, 32, 56, 48, 96, 22, 15, 31, 60, 51, 90, 124, 120, 83, 13, 39, 28, 91, 78, 84, 104, 195, 55, 18, 42, 72, 90, 48, 168, 144, 186, 234, 112, 12, 36, 48, 84, 72, 144, 96, 180, 156
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

This is the lower triangular array of A216626, which is the main entry for this sequence.

Examples

			The first rows of the triangle are:
1;
3,   7;
4,  12, 10;
7,  15, 28, 27;
6,  18, 24, 42, 16;
12, 28, 30, 60, 72,  70;
8,  24, 32, 56, 48,  96,  22;
15, 31, 60, 51, 90, 124, 120,  83;
13, 39, 28, 91, 78,  84, 104, 195, 55;

Links

Crossrefs

Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216626.

Programs

  • Maple
    with(numtheory):
T:= (n, k) -> add(add(ilcm(c, d), c=divisors(n)), d=divisors(k));
seq (seq (T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    T[n_, k_] := Sum[LCM[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)
  • Sage
    for n in (1..9): [A216626(n,k) for k in (1..n)]

Formula

T(n,1) = A000203(n) = sigma(n).
T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).

A062369 Dirichlet convolution of n and tau^2(n).

Original entry on oeis.org

1, 6, 7, 21, 9, 42, 11, 58, 30, 54, 15, 147, 17, 66, 63, 141, 21, 180, 23, 189, 77, 90, 27, 406, 54, 102, 106, 231, 33, 378, 35, 318, 105, 126, 99, 630, 41, 138, 119, 522, 45, 462, 47, 315, 270, 162, 51, 987, 86, 324, 147, 357, 57, 636, 135, 638, 161, 198, 63, 1323
Offset: 1

Views

Author

Vladeta Jovovic, Jul 07 2001

Keywords

Comments

Dirichlet convolution of A000027 and A035116.
Inverse Mobius transform of A060724. - R. J. Mathar, Oct 15 2011

Links

Crossrefs

Cf. A000203, A060648.
Cf. A000005, A060724, A064950, A062369, A062368, A062380.

Programs

  • Magma
    [&+[d*#Divisors(Floor(n/d))^2:d in Divisors(n)]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019
  • Mathematica
    a[n_] := Sum[ DivisorSigma[1, i]*DivisorSigma[1, j] / DivisorSigma[1, LCM[i, j]], {i, Divisors[n]}, {j, Divisors[n]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 26 2013 *)
  • PARI
    a(n) = sumdiv(n, d, d*numdiv(n/d)^2); \\ Michel Marcus, Nov 03 2018

Formula

a(n) = Sum_{i|n, j|n} sigma(i)*sigma(j)/sigma(lcm(i,j)), where sigma(n) = sum of divisors of n.
a(n) = Sum_{i|d, j|d} sigma(gcd(i, j));
a(n) = Sum_{d|n} d*tau(n/d)^2, where tau(n) = number of divisors of n.
Multiplicative with a(p^e) = (1-p^(3+e)-p^(2+e)+e^2+4*p^2+p^2*e^2+2*e-3*p+4*p^2*e-6*e*p-2*e^2*p)/(1-p)^3.
Dirichlet g.f.: (zeta(s))^4*zeta(s-1)/zeta(2*s). - R. J. Mathar, Feb 09 2011
G.f.: Sum_{k>=1} tau(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Sum_{k=1..n} a(k) ~ 5 * Pi^4 * n^2 / 144. - Vaclav Kotesovec, Jan 28 2019
a(n) = Sum_{d|n} tau(d^2)*sigma(n/d), where tau(n) = number of divisors of n, and sigma(n) = sum of divisors of n. - Ridouane Oudra, Aug 25 2019

A344133 a(n) = Sum_{i|n, j|n, k|n} i*j*k/gcd(i,j,k).

Original entry on oeis.org

1, 23, 46, 219, 116, 1058, 218, 1507, 883, 2668, 518, 10074, 716, 5014, 5336, 8819, 1208, 20309, 1502, 25404, 10028, 11914, 2186, 69322, 5691, 16468, 12628, 47742, 3452, 122728, 3938, 46995, 23828, 27784, 25288, 193377, 5588, 34546, 32936, 174812, 6848, 230644, 7526, 113442, 102428, 50278, 8978
Offset: 1

Views

Author

Seiichi Manyama, May 10 2021

Keywords

Crossrefs

Cf. A064950, A344132, A344134, A344135.

Programs

  • Mathematica
    a[n_]:= Sum[i*j*k/GCD[i,j,k], {i, (d = Divisors[n])}, {j, d}, {k, d}]; Array[a, 50] (* Amiram Eldar, May 10 2021 *)
  • PARI
    a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, i*j*k/gcd([i, j, k]))));

Formula

If p is prime, a(p) = 1 + 3*p + 4*p^2.

A344134 a(n) = Sum_{i|n, j|n, k|n} lcm(i,j,k).

Original entry on oeis.org

1, 15, 22, 91, 36, 330, 50, 387, 193, 540, 78, 2002, 92, 750, 792, 1363, 120, 2895, 134, 3276, 1100, 1170, 162, 8514, 511, 1380, 1192, 4550, 204, 11880, 218, 4275, 1716, 1800, 1800, 17563, 260, 2010, 2024, 13932, 288, 16500, 302, 7098, 6948, 2430, 330, 29986, 981, 7665, 2640, 8372, 372, 17880
Offset: 1

Views

Author

Seiichi Manyama, May 10 2021

Keywords

Crossrefs

Cf. A064950, A344132, A344133, A344135.

Programs

  • Mathematica
    a[n_]:= Sum[n/GCD[i,j,k], {i, (d = Divisors[n])}, {j, d}, {k, d}]; Array[a, 50] (* Amiram Eldar, May 10 2021 *)
  • PARI
    a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, lcm([i, j, k]))));
  • PARI
    a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, n/gcd([i, j, k]))));

Formula

a(n) = Sum_{i|n, j|n, k|n} n/gcd(i,j,k).
If p is prime, a(p) = 1 + 7*p.

A068984 a(n) = Sum_{d|n} d*tau(d)^2.

Original entry on oeis.org

1, 9, 13, 45, 21, 117, 29, 173, 94, 189, 45, 585, 53, 261, 273, 573, 69, 846, 77, 945, 377, 405, 93, 2249, 246, 477, 526, 1305, 117, 2457, 125, 1725, 585, 621, 609, 4230, 149, 693, 689, 3633, 165, 3393, 173, 2025, 1974, 837, 189, 7449, 470, 2214, 897
Offset: 1

Views

Author

Vladeta Jovovic, Apr 01 2002

Keywords

Links

Crossrefs

Cf. A000005, A062369, A060724, A064950.

Programs

  • Magma
    [&+[d*#Divisors(d)^2: d in Divisors(n)]:n in [1..51]]; // Marius A. Burtea, Sep 15 2019
  • Mathematica
    a[n_] := DivisorSum[n, # * DivisorSigma[0, #]^2 &]; Array[a, 100] (* Amiram Eldar, Sep 15 2019 *)
  • PARI
    a(n) = sumdiv(n, d, d*numdiv(d)^2) \\ Michel Marcus, Jun 17 2013

Formula

Multiplicative with a(p^e) = (p^(e+3)-3*p^(e+2)+4*p^(e+1)-p-1+2*p^(e+3)*e-6*p^(e+2)*e+4*p^(e+1)*e+p^(e+3)*e^2-2*p^(e+2)*e^2+p^(e+1)*e^2)/(p-1)^3.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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