A054525 -id:A054525 - cp's OEIS frontend

A143153 Triangle read by rows, A054525 * (A020639 * 0^(n-k)), 1<=k<=n.

1, -1, 2, -1, 0, 3, 0, -2, 0, 2, -1, 0, 0, 0, 5, 1, -2, -3, 0, 0, 2, -1, 0, 0, 0, 0, 0, 7, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -3, 0, 0, 0, 0, 0, 3, 1, -2, 0, 0, -5, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gary W. Adamson & Mats Granvik, Jul 27 2008

Right border = A020639, Lpf(n): (1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11,...).
Left border = mu(n), A008683: (1, -1, -1, 0, -1, 1, -1,...).
Row sums = a signed version of A097945: (1, -1, -2, 0, -4, 2, -6,...) such that parity = (+) iff mu(n) = +.

			First few rows of the triangle =
1;
-1, 2;
-1, 0, 3;
0, -2, 0, 2;
-1, 0, 0, 0, 5;
1, -2, -3, 0, 0, 2;
-1, 0, 0, 0, 0, 0, 7;
0, 0, 0, -2, 0, 0, 0, 2;
0, 0, -3, 0, 0, 0, 0, 0, 3;
1, -2, 0, 0, -5, 0, 0, 0, 0, 2;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 2;
...

Cf. A008683, A097945, A020639, A054525.

Triangle read by rows, A054525 * (A020639 * 0^(n-k)), 1<=k<=n; where A020639 = Lpf(n): (1, 2, 3, 2, 5, 2, 7, 2, 3, 2,...) and A054525 = the Mobius transform.

A143258 Triangle read by rows, A054525 * A143257, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 6, 2, 2, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Aug 02 2008

Left border = phi(n), A000010.

Row sums = A029935: (1, 2, 4, 5, 8, 8, 12, 12,...)

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
4, 1, 1, 1, 1;
2, 2, 1, 1, 1, 1;
6, 1, 1, 1, 1, 1, 1;
...

Cf. A143257, A054525, A029935, A000010.

Mobius transform of triangle A143257

A143275 A054525 * A029935.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 11, 7, 12, 7, 19, 9, 23, 11, 21, 16, 31, 12, 35, 21, 33, 19, 43, 21, 48, 23, 44, 33, 55, 21, 59, 36, 57, 31, 77, 36, 71, 35, 69, 49, 79, 33, 83, 57, 84, 43, 91, 48, 108, 48, 93, 69, 103, 44, 133, 77, 105, 55, 115, 63, 119, 59, 132, 80, 161, 57, 131, 93, 129

Offset: 1

Gary W. Adamson, Aug 03 2008

			a(4) = 3 = (0, -1, 0, 1) dot (1, 2, 4, 5) = (0 - 2 + 0 + 5), where K(0, -1, 0, 1) = row 4 of A054525 and A143275 = (1, 2, 4, 5, 8, 8, 12, ...).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A029935, A054525.

read("transforms") : A029935 := proc(n) local a,d ; a := 0 ; for d in numtheory[divisors](n) do a := a+ numtheory[phi](d)*numtheory[phi](n/d); od; RETURN(a) ; end: a029935 := [seq(A029935(n),n=1..300)] ; a143275 := MOBIUS(a029935) ; # R. J. Mathar, Jan 19 2009

f[p_, e_] := If[e > 1, (e*(p-1) + p + 2) * (p-1)^2 * p^(e-3), 2*p - 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)

Möbius transform (A054525) of A029935: (1, 2, 4, 5, 8, 8, 12, 12, 16, ...).

Multiplicative with a(p) = 2*p - 3 and a(p^e) = (e*(p-1) + p + 2) * (p-1)^2 * p^(e-3) for e > 1. - Amiram Eldar, Aug 31 2023

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

A143276 Triangle read by rows: A054525 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 5, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...).

Left border = A007431: (1, 0, 1, 1, 3, 0, 5, 2, 4,...).

			First few rows of the triangle =
1;
0, 1;
1, 0, 1;
1, 0, 0, 1;
3, 0, 0, 0, 1;
0, 1, 0, 0, 0, 1;
5, 0, 0, 0, 0, 0, 1;
2, 1, 0, 0, 0, 0, 0, 1;
...

Cf. A000010, A007431, A054523, A054525.

Moebius transform of triangle A054523.

a(17), a(18) corrected by Georg Fischer, May 29 2023

A143309 Triangle read by rows, A054525 * A143308, 1<=k<=n.

Original entry on oeis.org

1, 3, 2, 5, 3, 3, 8, 6, 4, 4, 9, 5, 5, 5, 5, 15, 13, 9, 6, 6, 6, 13, 7, 7, 7, 7, 7, 7, 20, 16, 12, 12, 8, 8, 8, 8, 21, 15, 15, 9, 9, 9, 9, 9, 9, 27, 23, 15, 15, 15, 10, 10, 10, 10, 10, 21, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 40, 36, 32, 26, 18, 18, 12, 12, 12, 12, 12, 12, 25, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = A069097: (1, 5, 11, 22, 29, 55,...).

Left border = A018804: (1, 3, 5, 8, 9, 15,...).

			First few rows of the triangle =
1;
3, 2;
5, 3, 3;
8, 6, 4, 4;
9, 5, 5, 5, 5;
15, 13, 9, 6, 6, 6;
13, 7, 7, 7, 7, 7, 7;
...

Cf. A018804, A069097, A143308.

Mobius transform of triangle A143308.

a(56) = 10 removed and more terms from Georg Fischer, Jul 13 2023

A143312 Triangle read by rows, A054525 * A143311, 1<=k<=n.

Original entry on oeis.org

1, 3, 2, 8, 0, 3, 12, 6, 0, 4, 24, 0, 0, 0, 5, 24, 16, 9, 0, 0, 6, 48, 0, 0, 0, 0, 0, 7, 48, 24, 0, 12, 0, 0, 0, 8, 72, 0, 24, 0, 0, 0, 0, 0, 9, 72, 48, 0, 0, 15, 0, 0, 0, 0, 10, 120, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 96, 48, 36, 32, 0, 18, 0, 0, 0, 0, 0, 12, 168, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Gary W. Adamson, Aug 06 2008

Left border = A007434.

Row sums = A069097: (1, 5, 11, 2, 29, 55,...)

			First few rows of the triangle =
1;
3, 2;
8, 0, 3;
12, 6, 0, 4;
24, 0, 0, 0, 5;
24, 16, 9, 0, 0, 6;
48, 0, 0, 0, 0, 0, 7;
48, 24, 0, 12, 0, 0, 0, 8;
...

Cf. A054525, A143311, A007434, A069097.

Mobius transform of triangle A143311

A143316 Triangle read by rows, A054525 * A143315, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 4, 2, 0, 1, 8, 0, 0, 0, 1, 4, 4, 2, 0, 0, 1, 12, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 12, 0, 4, 0, 0, 0, 0, 0, 1, 8, 8, 0, 0, 2, 0, 0, 0, 0, 1, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 4, 4, 4, 0, 2, 0, 0, 0, 0, 0, 1, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 12, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = 2*n-1.

Left border = A140434: (1, 2, 4, 4, 8, 4, 12, 8,...).

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

Cf. A054525, A140434, A143315.

a(19) corrected and more terms from Georg Fischer, Aug 14 2023

A143444 A054525 * the primes prefaced by 0.

Original entry on oeis.org

0, 2, 3, 3, 7, 6, 13, 12, 16, 14, 29, 17, 37, 26, 33, 30, 53, 32, 61, 41, 55, 42, 79, 40, 82, 58, 82, 59, 107, 44, 113, 80, 99, 82, 119, 70, 151, 94, 123, 88, 173, 74, 181, 115, 134, 116, 199, 98, 210, 122, 173, 133, 239, 100, 215, 142, 199, 160, 271, 107, 281, 168, 206

Offset: 1

Gary W. Adamson, Aug 15 2008

nonn

a(5) = 7 = (-1, 0, 0, 0, 1) dot (0, 2, 3, 5, 7).

a(p(n)) = p(n-1) since prime rows of the Mobius transform have only one nonzero multiplier besides the (-1) which is multiplied * the 0 in (0, 2, 3, 5, 7,...). Cf. A055003: (1, 3, 7, 13, 29, 37,...) a subset of primes in the sequences, but other terms may be primes.

			a(6) = 6 = (1, -1, -1, 0, 0, 1) dot (0, 2, 3, 5, 7, 11) where (1, -1, -1, 0, 0, 1) = row 6 of A051731, the Mobius transform.

Cf. A000040, A051731, A055003.

Mobius transform of the primes prefaced by 0.

a(12) replaced by 17 and extended by R. J. Mathar, Jan 19 2009

A143517 Triangle read by rows, A054525 * (A061397 * 0^(n-k)), 1<=k<=n.

Original entry on oeis.org

0, 0, 2, 0, 0, 3, 0, -2, 0, 0, 0, 0, 0, 0, 5, 0, -2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Gary W. Adamson, Aug 22 2008

Row sums = A137851: (0, 2, 3, -2, 5, -5, 7, 0, -3,...).

Right border = A061397: (0, 2, 3, 0, 5, 0, 7,...).

			First few rows of the triangle =
0;
0, 2;
0, 0, 3;
0, -2, 0, 0;
0, 0, 0, 0, 5;
0, -2, -3, 0, 0, 0;
0, 0, 0, 0, 0, 0, 7;
...

Cf. A054525, A061397, A137851.

Triangle read by rows, A054525 * (A061397 * 0^(n-k)), 1<=k<=n

A143518 Triangle read by rows, A054525 * (A010051 * 0^(n-k)), 1<=k<=n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 22 2008

Row sums = A143519: (0, 1, 1, -1, 1, -2, 1, 0, -1,...)

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

Cf. A010051, A054525.

Triangle read by rows, A054525 * A010051, 1<=k<=n. A054525 = the Mobius transform. A010051 * 0^(n-k) = an infinite lower triangular matrix with A010051, the characteristic function of the primes (0, 1, 1, 0, 1, 0, 1,...) as the main diagonal and the rest zeros.

cp's OEIS Frontend

A143153 Triangle read by rows, A054525 * (A020639 * 0^(n-k)), 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143258 Triangle read by rows, A054525 * A143257, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143275 A054525 * A029935.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A143276 Triangle read by rows: A054525 * A054523 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A143309 Triangle read by rows, A054525 * A143308, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A143312 Triangle read by rows, A054525 * A143311, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143316 Triangle read by rows, A054525 * A143315, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A143444 A054525 * the primes prefaced by 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A143517 Triangle read by rows, A054525 * (A061397 * 0^(n-k)), 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula