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.

A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 3, 4, 4, 3, 2, 8, 6, 8, 2, 4, 4, 6, 6, 4, 4, 2, 10, 4, 15, 4, 10, 2, 4, 4, 12, 6, 6, 12, 4, 4, 3, 11, 4, 16, 8, 16, 4, 11, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 10, 10, 22, 4, 30, 4, 22, 10, 10, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

T(n,k) = number of subgroups of C_n X C_k. [Hampjes et al.] - N. J. A. Sloane, Feb 02 2013

Examples

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  5,  4,  8,  4, 10,  4, 11,  6, 10,  4, 16
[ 3] 2,  4,  6,  6,  4, 12,  4,  8, 10,  8,  4, 18
[ 4] 3,  8,  6, 15,  6, 16,  6, 22,  9, 16,  6, 30
[ 5] 2,  4,  4,  6,  8,  8,  4,  8,  6, 16,  4, 12
[ 6] 4, 10, 12, 16,  8, 30,  8, 22, 20, 20,  8, 48
[ 7] 2,  4,  4,  6,  4,  8, 10,  8,  6,  8,  4, 12
[ 8] 4, 11,  8, 22,  8, 22,  8, 37, 12, 22,  8, 44
[ 9] 3,  6, 10,  9,  6, 20,  6, 12, 23, 12,  6, 30
[10] 4, 10,  8, 16, 16, 20,  8, 22, 12, 40,  8, 32
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 14, 12
[12] 6, 16, 18, 30, 12, 48, 12, 44, 30, 32, 12, 90
.
Displayed as a triangular array:
1,
2,  2,
2,  5,  2,
3,  4,  4,  3,
2,  8,  6,  8, 2,
4,  4,  6,  6, 4,  4,
2, 10,  4, 15, 4, 10, 2,
4,  4, 12,  6, 6, 12, 4,  4,
3, 11,  4, 16, 8, 16, 4, 11, 3,

Links

Crossrefs

Cf. A216620, A216621, A216622, A216623, A216625, A216626, A216627.
Main diagonal is A060724.
Rows give A000005, A060710, A054584, A221951, A221852.

Programs

  • Maple
    with(numtheory):
T:= (n, k)-> add(add(igcd(c,d), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14); # Alois P. Heinz, Sep 12 2012
T:=proc(m,n) local d; add( d*tau(m*n/d^2), d in divisors(gcd(m,n))); end; # N. J. A. Sloane, Feb 02 2013
  • Mathematica
    t[n_, k_] := Sum[Sum[GCD[c, d], {c, Divisors[n]}], {d, Divisors[k]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 21 2013 *)
  • Sage
    def A216624(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(gcd, cp))
for n in (1..12): [A216624(n,k) for k in (1..12)]

Formula

T(n,n) = A060724(n) = sum_{d|n} d*tau((n/d)^2).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A060710(n) = sum_{d|n} (3-[d is odd]) (Iverson bracket).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A113935(n) = prime(n)+3.

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)]

A216620 Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 3, 4, 4, 3, 2, 6, 5, 6, 2, 4, 4, 6, 6, 4, 4, 2, 8, 4, 10, 4, 8, 2, 4, 4, 10, 6, 6, 10, 4, 4, 3, 8, 4, 12, 7, 12, 4, 8, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 8, 8, 14, 4, 20, 4, 14, 8, 8, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6, 2, 12, 4, 12, 6
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

T(n,n) = A060648(n) = Sum_{d|n} Dedekind_Psi(d).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A062011(n) = 2*tau(n).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A052147(n) = prime(n)+2.

Examples

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8,  4, 12
[ 3] 2,  4,  5,  6,  4, 10,  4,  8,  8,  8,  4, 15
[ 4] 3,  6,  6, 10,  6, 12,  6, 14,  9, 12,  6, 20
[ 5] 2,  4,  4,  6,  7,  8,  4,  8,  6, 14,  4, 12
[ 6] 4,  8, 10, 12,  8, 20,  8, 16, 16, 16,  8, 30
[ 7] 2,  4,  4,  6,  4,  8,  9,  8,  6,  8,  4, 12
[ 8] 4,  8,  8, 14,  8, 16,  8, 22, 12, 16,  8, 28
[ 9] 3,  6,  8,  9,  6, 16,  6, 12, 17, 12,  6, 24
[10] 4,  8,  8, 12, 14, 16,  8, 16, 12, 28,  8, 24
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 13, 12
[12] 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12, 50
.
Displayed as a triangular array:
   1,
   2, 2,
   2, 4,  2,
   3, 4,  4,  3,
   2, 6,  5,  6, 2,
   4, 4,  6,  6, 4,  4,
   2, 8,  4, 10, 4,  8, 2,
   4, 4, 10,  6, 6, 10, 4, 4,
   3, 8,  4, 12, 7, 12, 4, 8, 3,

Links

Crossrefs

Cf. A216621, A216622, A216623, A216624, A216625, A216626, A216627.

Programs

  • Maple
    with(numtheory):
T:= (n, k)-> add(add(phi(igcd(c,d)), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    t[n_, k_] := Outer[ EulerPhi[ GCD[#1, #2]]&, Divisors[n], Divisors[k]] // Flatten // Total; Table[ t[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)
  • Sage
    def A216620(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(euler_phi, map(gcd, cp)))
for n in (1..12): [A216620(n,k) for k in (1..12)]

A216622 Square array read by antidiagonals: T(n,k) = Sum_{c|n, d|k} phi(lcm(c,d)) for n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 7, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 14, 15, 12, 7, 8, 14, 14, 20, 20, 14, 14, 8, 9, 16, 21, 24, 13, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 19, 26, 35, 28, 35, 26, 19, 20, 11, 12, 22, 30, 36, 40
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

T(n,n) = A062380(n) = Sum_{d|n} phi(d)*tau(d^2).
T(n,1) = T(1,n) = A000027(n) = n.
T(n,2) = T(2,n) = A005843(n) = 2*n.
T(n+1,n) = A002378(n) = (n+1)*n.
T(prime(n),1) = A000040(n) = prime(n).
T(prime(n),prime(n)) = 3*prime(n)-2.

Examples

			[-----1---2---3---4---5---6---7---8---9---10---11---12]
[ 1]  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12
[ 2]  2,  4,  6,  8, 10, 12, 14, 16, 18,  20,  22,  24
[ 3]  3,  6,  7, 12, 15, 14, 21, 24, 19,  30,  33,  28
[ 4]  4,  8, 12, 14, 20, 24, 28, 26, 36,  40,  44,  42
[ 5]  5, 10, 15, 20, 13, 30, 35, 40, 45,  26,  55,  60
[ 6]  6, 12, 14, 24, 30, 28, 42, 48, 38,  60,  66,  56
[ 7]  7, 14, 21, 28, 35, 42, 19, 56, 63,  70,  77,  84
[ 8]  8, 16, 24, 26, 40, 48, 56, 42, 72,  80,  88,  78
[ 9]  9, 18, 19, 36, 45, 38, 63, 72, 37,  90,  99,  76
[10] 10, 20, 30, 40, 26, 60, 70, 80, 90,  52, 110, 120
[11] 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,  31, 132
[12] 12, 24, 28, 42, 60, 56, 84, 78, 76, 120, 132,  98
.
Displayed as a triangular array:
   1,
   2,  2,
   3,  4,  3,
   4,  6,  6,  4,
   5,  8,  7,  8,  5,
   6, 10, 12, 12, 10,  6,
   7, 12, 15, 14, 15, 12,  7,
   8, 14, 14, 20, 20, 14, 14,  8,
   9, 16, 21, 24, 13, 24, 21, 16,  9,

Links

Crossrefs

Cf. A216620, A216621, A216623, A216624, A216625, A216626, A216627.

Programs

  • Maple
    with(numtheory):
T:= (n, k)-> add(add(phi(ilcm(c, d)), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    t[n_, k_] := Sum[ EulerPhi[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, Jun 28 2013 *)
  • Sage
    def A216622(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
for n in (1..12): [A216622(n,k) for k in (1..12)]

A216623 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = Sum_{c|n,d|k} phi(lcm(c,d)).

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 4, 8, 12, 14, 5, 10, 15, 20, 13, 6, 12, 14, 24, 30, 28, 7, 14, 21, 28, 35, 42, 19, 8, 16, 24, 26, 40, 48, 56, 42, 9, 18, 19, 36, 45, 38, 63, 72, 37, 10, 20, 30, 40, 26, 60, 70, 80, 90, 52, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 31, 12, 24
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

This is the lower triangular array of A216622, which is the main entry for this sequence.
T(n,1) = A000027(n).
T(n,n) = A062380(n).

Examples

			The first rows of the triangle are:
1,
2,  4,
3,  6,  7,
4,  8, 12, 14,
5, 10, 15, 20, 13,
6, 12, 14, 24, 30, 28,
7, 14, 21, 28, 35, 42, 19,
8, 16, 24, 26, 40, 48, 56, 42,
9, 18, 19, 36, 45, 38, 63, 72, 37,

Links

Crossrefs

Cf. A216620, A216621, A216622, A216624, A216625, A216626, A216627.

Programs

  • Maple
    with(numtheory):
T:= (n, k)-> add(add(phi(ilcm(c, d)), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    t[n_, k_] := Sum[ EulerPhi[ LCM[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 23 2013 *)
  • Sage
    # uses[A216622]
for n in (1..9): [A216622(n,k) for k in (1..n)]

A216625 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} gcd(c,d).

Original entry on oeis.org

1, 2, 5, 2, 4, 6, 3, 8, 6, 15, 2, 4, 4, 6, 8, 4, 10, 12, 16, 8, 30, 2, 4, 4, 6, 4, 8, 10, 4, 11, 8, 22, 8, 22, 8, 37, 3, 6, 10, 9, 6, 20, 6, 12, 23, 4, 10, 8, 16, 16, 20, 8, 22, 12, 40, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 14, 6, 16, 18, 30, 12, 48, 12, 44, 30, 32
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

This is the lower triangular array of A216624, which is the main entry for this sequence.
T(n,1) = A000005(n) = tau(n).
T(n,n) = A060724(n) = Sum_{d|n} d*tau((n/d)^2).

Examples

			The first rows of the triangle are:
  1;
  2,  5;
  2,  4,  6;
  3,  8,  6, 15;
  2,  4,  4,  6,  8;
  4, 10, 12, 16,  8, 30;
  2,  4,  4,  6,  4,  8, 10;
  4, 11,  8, 22,  8, 22,  8, 37;
  3,  6, 10,  9,  6, 20,  6, 12, 23;

Links

Crossrefs

Cf. A216620, A216621, A216622, A216623, A216624, A216626, A216627.

Programs

  • Maple
    with(numtheory):
T:= (n, k)-> add(add(igcd(c, d), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    T[n_, k_] := Sum[GCD[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)
  • Sage
    for n in (1..9): [A216624(n,k) for k in (1..n)]

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).
Showing 1-7 of 7 results.

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