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.

Previous Showing 21-30 of 99 results. Next

A143424 Triangle read by rows, A054525 * A130123, 1<=k<=n.

Original entry on oeis.org

1, -1, 2, -1, 0, 4, 0, -2, 0, 8, -1, 0, 0, 0, 16, 1, -2, -4, 0, 0, 32, -1, 0, 0, 0, 0, 0, 64, 0, 0, 0, -8, 0, 0, 0, 128, 0, 0, -4, 0, 0, 0, 0, 0, 256, 1, -2, 0, 0, -16, 0, 0, 0, 0, 512, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 2, 0, -8, 0, -32, 0, 0, 0, 0, 0, 2048
Offset: 1

Views

Author

Gary W. Adamson, Aug 14 2008

Keywords

Comments

Row sums = A000740 starting with offset 1: (1, 1, 3, 6, 15, 27, 63,...).
Left border = mu(n), A008683.

Examples

			First few rows of the triangle =
1;
-1, 2;
-1, 0, 4;
0, -2, 0, 8;
-1, 0, 0, 0, 16;
1, -2, -4, 0, 0, 32;
-1, 0, 0, 0, 0, 0, 64;
0, 0, 0, -8, 0, 0, 0, 128;
...

Crossrefs

Cf. A000740, A130123, A054525, A008683.

Formula

Mobius transform of A130123, where A130123 = an infinite lower triangular matrix with (1, 2, 4, 8,...) in the main diagonal and the rest zeros.

A159937 Triangle read by rows, A054525 * A127478, as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 2, 1, 0, 2, 4, 0, 0, 0, 4, 2, 2, 2, 0, 0, 2, 6, 0, 0, 0, 0, 0, 6, 4, 2, 0, 2, 0, 0, 0, 4, 6, 0, 4, 0, 0, 0, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 4, 2, 4, 4, 0, 2, 0, 0, 0, 0, 0, 4
Offset: 1

Views

Author

Gary W. Adamson, Apr 26 2009

Keywords

Comments

Row sums = A029935: (1, 2, 4, 5, 8, 8,...). Right and left borders = A000010, phi(n).

Examples

			First few rows of the triangle =
1;
1, 1;
2, 0, 2;
2, 1, 0, 2;
4, 0, 0, 0, 4;
2, 2, 2, 0, 0, 2;
6, 0, 0, 0, 0, 0, 6;
4, 2, 0, 2, 0, 0, 0, 4;
6, 0, 4, 0, 0, 0, 0, 0, 6;
4, 4, 0, 0, 4, 0, 0, 0, 0, 4;
10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
4, 2, 4, 4, 0, 2, 0, 0, 0, 0, 0, 4;
12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12;
6, 6, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 6;
8, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8;
...

Crossrefs

Cf. A127478, A029935

Formula

A054525 * A127478 = Mobius transform of triangle A127478.

A127627 Triangle T(n,k) = A054525(n,k)*A018804(k), read by rows 1<=k<=n.

Original entry on oeis.org

1, -1, 3, -1, 0, 5, 0, -3, 0, 8, -1, 0, 0, 0, 9, 1, -3, -5, 0, 0, 15, -1, 0, 0, 0, 0, 0, 13, 0, 0, 0, -8, 0, 0, 0, 20, 0, 0, -5, 0, 0, 0, 0, 0, 21, 1, -3, 0, 0, -9, 0, 0, 0, 0, 27, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 3, 0, -8, 0, -15, 0, 0, 0, 0, 0, 40
Offset: 1

Views

Author

Gary W. Adamson, Jan 20 2007

Keywords

Examples

			First few rows of the triangle are:
1;
-1, 3;
-1, 0, 5;
0, -3, 0, 8;
-1, 0, 0, 0, 9;
1, -3,-5, 0, 0, 15;
...

Crossrefs

Cf. A054525, A018804, A029935 (row sums).

Programs

Formula

T(n,1) = A008683(n).
T(n,n) = A018804(n).

A128521 A128174 * A054525 * A000012.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 0, 1, 2, 1, 1, 1, 3, 3, 2, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 1, 3, 3, 3, 3, 2, 2, 1, 1, 0, -1, 1, 2, 2, 3, 2, 2, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Mar 07 2007

Keywords

Comments

Row sums = A106477: (1, 1, 3, 3, 7, 5, 13, 9, 19, 13, ...). A128522 = A054525 * A128174 * A000012.

Examples

			First few rows of the triangle:
  1;
  0, 1;
  1, 1, 1;
  0, 1, 1, 1;
  1, 2, 2, 1, 1;
  0, 0, 1, 2, 1, 1;
  1, 3, 3, 2, 2, 1, 1;
  0, 0, 1, 2, 2, 2, 1, 1;
  ...

Crossrefs

Cf. A128174, A054525, A000012, A106477, A128522.

Formula

A128174 * A054525 * A000012 as infinite lower triangular matrices.

A128522 A054525 * A128174 * A000012.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 3, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 3, 2, 2, 1, 1, 2, 2, 2, 3, 2, 3, 2, 2, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Mar 07 2007

Keywords

Comments

Row sums = A123323: (1, 1, 3, 4, 8,7, 15, 14, ...). Left column = A083290: (1, 0, 1, 1, 2, 1, 3, 2, 3, 2, ...) A128521 = A128174 * A054525 * A000012.

Examples

			First few rows of the triangle:
  1;
  0, 1;
  1, 1, 1;
  1, 1, 1, 1;
  2, 2, 2, 1, 1;
  1, 1, 1, 2, 1, 1;
  3, 3, 3, 2, 2, 1, 1;
  2, 2, 2, 2, 2, 2, 1, 1;
  3, 3, 3, 3, 3, 2, 2, 1, 1;
  ...

Crossrefs

Cf. A054525, A128174, A000012, A083290, A123323, A128521.

Formula

A054525 * A128174 * A000012 as infinite lower triangular matrices.

A130159 A054525 * A000069.

Original entry on oeis.org

1, 1, 3, 5, 7, 6, 12, 7, 12, 10, 20, 6, 24, 12, 17, 17, 31, 12, 36, 14, 25, 20, 43, 18, 41, 24, 36, 24, 55, 14, 60, 31, 40, 34, 49, 24, 72, 36, 48, 34, 80, 22, 83, 40, 48, 46, 92, 30, 84, 38, 65, 48, 103, 36, 81, 48, 72
Offset: 1

Views

Author

Gary W. Adamson, May 13 2007

Keywords

Examples

			a(4) = 5 = dot product of row 4 of A054525, [0, -1, 0, 1] and [1, 2, 4, 7], where A000069 = (1, 2, 4, 7, 8, 11, 13, ...).

Crossrefs

Cf. A130160, A000069.

Formula

Moebius transform of A000069.

Extensions

Extended by R. J. Mathar, Apr 04 2012

A133703 A054525 * A133701.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Gary W. Adamson, Sep 21 2007

Keywords

Comments

Right border = A001227: (1, 1, 2, 1, 2, 2, 2, 1, 3, ...), the number of odd divisors of n.
Row sums = A023136: (1, 1, 3, 1, 3, 3, 3, 1, 5, 3, ...).

Examples

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 2;
  0, 0, 0, 1;
  1, 0, 0, 0, 2;
  0, 1, 0, 0, 0, 2;
  1, 0, 0, 0, 0, 0, 2;
  ...

Crossrefs

Cf. A054525, A133701, A001227, A023136.

Formula

Mobius transform of triangle A133701.

A134764 A054525 * A000984.

Original entry on oeis.org

1, 1, 5, 18, 69, 245, 923, 3412, 12864, 48549, 184755, 705162, 2704155, 10399675, 40116525, 155114088, 601080389, 2333593104, 9075135299, 35345215162, 137846527891, 538257689683, 2104098963719, 8233430018756, 32247603683030
Offset: 1

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Examples

			a(4) = 18 = (0, -1, 0, 1) dot (1, 2, 6, 20), where A000984 = (1, 2, 6, 20, 70, 252, ...) and (0, -1, 0, 1) = row 4 of triangle A054525.

Crossrefs

Cf. A000984, A054525.

Programs

  • Maple
    read("transforms") : A000984 := proc(n) binomial(2*n,n) ; end: a000984 := [seq(A000984(n),n=0..50)] ; a134764 := MOBIUS(a000984) ; for i from 1 to nops(a134764) do printf("%d,",op(i,a134764)) ; od: # R. J. Mathar, Jan 19 2009

Formula

Möbius transform of A000984.

Extensions

More terms from R. J. Mathar, Jan 19 2009

A134842 Triangle, n-th row = first n terms of n-th row of an array formed by A054525 * A051731(transform).

Original entry on oeis.org

1, -1, 0, -1, -1, 0, 0, -1, 0, 0, -1, -1, -1, -1, 0, 1, 0, 0, 0, 1, 0, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0
Offset: 1

Views

Author

Gary W. Adamson, Nov 12 2007

Keywords

Comments

Left column = mu(n), A008683: (1, -1, -1, 0, -1, 1, 0, 0, 1, ...).
Row sums = A023900: (1, -1, -2, -1, -4, 2, -6, ...).

Examples

			First few rows of the array:
   1,  1,  1,  1,  1,  1,  1, ...
  -1,  0, -1,  0, -1,  0, -1, ...
  -1, -1,  0, -1, -1, -1,  0, ...
   0, -1,  0,  0,  0, -1,  0, ...
  -1, -1, -1, -1, -1,  0, -1, ...
  ...
First few rows of the triangle:
   1;
  -1,  0;
  -1, -1,  0;
   0, -1,  0,  0;
  -1, -1, -1, -1,  0;
   1,  0,  0,  0,  1,  0;
  -1, -1, -1, -1, -1, -1,  0;
  ...

Crossrefs

Cf. A051731, A054525, A008683, A023900.

Formula

Triangle, n-th row - first n terms of n-th row of an array formed by A054525 * A051731(transform).

A140706 A054525 * A014683; a(n) = Sum_{d|n} mu(d)*A014683(n/d).

Original entry on oeis.org

1, 2, 3, 1, 5, 0, 7, 4, 5, 2, 11, 5, 13, 4, 6, 8, 17, 7, 19, 9, 10, 8, 23, 8, 19, 10, 18, 13, 29, 11, 31, 16, 18, 14, 22, 12, 37, 16, 22, 16, 41, 15, 43, 21, 25, 20, 47, 16, 41, 21, 30, 25, 53, 18, 38, 24, 34, 26, 59, 15, 61, 28, 37, 32, 46, 23, 67, 33, 42, 27, 71, 24, 73, 34, 41
Offset: 1

Views

Author

Gary W. Adamson, May 24 2008

Keywords

Comments

a(n) = n iff n is prime.

Examples

			a(4) = 1 = (0, -1, 0, 1) dot (1, 3, 4, 4), where (0, -1, 0, 1) = row 4 of triangle A054525.

Links

Crossrefs

Cf. A008683, A014683, A054525.

Programs

  • Maple
    read("transforms") : A014683 := proc(n) if isprime(n) then 1+n; else n; fi; end: a014683 := [seq(A014683(n),n=1..150)] ; a140706 := MOBIUS(a014683) ; for i from 1 to nops(a140706) do printf("%d,",op(i,a140706)) ; od: # R. J. Mathar, Jan 19 2009
  • Mathematica
    Table[Sum[MoebiusMu[d] (# + Boole@ PrimeQ@ #) &[n/d], {d, Divisors@ n}], {n, 75}] (* Michael De Vlieger, Jul 29 2017 *)
  • PARI
    A014683(n) = (n+isprime(n));
A140706(n) = sumdiv(n,d,moebius(d)*A014683(n/d)); \\ Antti Karttunen, Jul 28 2017
  • Python
    from sympy import isprime, mobius, divisors
def a014683(n): return n + isprime(n)
def a140706(n): return sum(mobius(d)*a014683(n//d) for d in divisors(n))
print([a140706(n) for n in range(1,51)]) # Indranil Ghosh, Jul 29 2017

Formula

Möbius transform of A014683: (1, 3, 4, 4, 6, 6, 8, 8, 9, 10, ...); where A014683(n) = n if n is not prime; but (n+1) if n is prime.
a(n) = Sum_{d|n} A008683(d)*A014683(n/d), where A008683 is Moebius mu function. - Antti Karttunen, Jul 28 2017

Extensions

More terms from R. J. Mathar, Jan 19 2009
Second part added to the name by Antti Karttunen, Jul 28 2017
Previous Showing 21-30 of 99 results. Next

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

Content is available under The OEIS End-User License Agreement.