A077049 -id:A077049 - cp's OEIS frontend

A077052 Right Moebius transformation matrix, M, by antidiagonals.

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

Offset: 1

Clark Kimberling, Oct 22 2002

If S=(s(1),s(2),...) is a sequence written as a row vector, then S*M is the Moebius transform of S; i.e. its n-th term is Sum{mu(k)*s(k): k|n}. M is the transpose of the left Moebius transformation matrix, A077050.

			Northwest corner:
1 -1 -1 0 -1 1
0 1 0 -1 0 -1
0 0 1 0 0 -1
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1

Cf. A077049, A077050, A077051.

M=T^(-1), where T is the right summatory matrix, A077051.

A171233 Array, T(n,k) = 2*(n/k), if n mod k = 0; otherwise, T(n,k) = 1. Read by antidiagonals.

Original entry on oeis.org

2, 4, 1, 6, 2, 1, 8, 1, 1, 1, 10, 4, 2, 1, 1, 12, 1, 1, 1, 1, 1, 14, 6, 1, 2, 1, 1, 1, 16, 1, 4, 1, 1, 1, 1, 1, 18, 8, 1, 1, 2, 1, 1, 1, 1, 20, 1, 1, 1, 1, 1, 1, 1, 1, 1, 22, 10, 6, 4, 1, 2, 1, 1, 1, 1, 1, 24, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 26, 12, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 28, 1, 8, 1, 4, 1, 1, 1

Offset: 1

Ross La Haye, Dec 05 2009

T(n,3): continued fraction expansion of e - 1.

			Array begins
2 1 1 1 ...
4 2 1 1 ...
6 1 2 1 ...
8 4 1 2 ...
...........

Cf. T(n,1) = A005843(n-1), A171232, A077049.

A171233 := proc(n,k) if n mod k <> 0 then 1; else 2*n/k ; end if; end proc: seq(seq(A171233(d-k+1,k),k=1..d),d=1..17) ; # R. J. Mathar, Dec 08 2009

T(n,k) = A171232(n,k) + A077049(n,k).

Terms beyond the 6th antidiagonal from R. J. Mathar, Dec 08 2009

A247146 As a binary numeral, the bit 2^(m-1) of a(n) is 1 iff m is a proper divisor of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 11, 5, 19, 1, 47, 1, 67, 21, 139, 1, 295, 1, 539, 69, 1027, 1, 2223, 17, 4099, 261, 8267, 1, 16951, 1, 32907, 1029, 65539, 81, 133423, 1, 262147, 4101, 524955, 1, 1056871, 1, 2098187, 16661, 4194307, 1, 8423599, 65, 16777747, 65541

Offset: 1

Morgan L. Owens, Nov 21 2014

nonn

a(n)==1 iff n is prime.
Apparently Moebius transform of A178472.
For n>1, the binary representation of a(n) is given by row (n-1) of A077049 (when read as a triangular array). - Tom Edgar, Nov 28 2014

Cf. A034729, A077049, A178472.

With[{n=Range[100]},(1/2) ((Total/@(2^Divisors[n])) - 2^n)]

a(n) = sumdiv(n, k, 2^(k-1)) - 2^(n-1); \\ Michel Marcus, Nov 25 2014

from sympy import divisors
def A247146(n): return sum(1<Chai Wah Wu, Jul 15 2022

a(n) = A034729(n) - 2^(n-1). - Michel Marcus, Nov 22 2014

A171232 Array read by antidiagonals, T(n,k) = 2*(n/k) - 1, if n mod k = 0; otherwise, T(n,k) = 1.

Original entry on oeis.org

1, 3, 1, 5, 1, 1, 7, 1, 1, 1, 9, 3, 1, 1, 1, 11, 1, 1, 1, 1, 1, 13, 5, 1, 1, 1, 1, 1, 15, 1, 3, 1, 1, 1, 1, 1, 17, 7, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 9, 5, 3, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 11, 1, 1, 1, 1

Offset: 1

Ross La Haye, Dec 05 2009

T(n,1): continued fraction expansion of coth(1).

T(n,2): continued fraction expansion of tan(1) = cot(pi/2 - 1).

			Array begins
1 1 1 1 1 ...
3 1 1 1 1 ...
5 1 1 1 1 ...
7 3 1 1 1 ...
9 1 1 1 1 ...
.............

Cf. T(n, 1) = A005408(n-1), T(n, 2) = A093178(n-1), A171233, A077049.

T[n_,k_] := If[Divisible[n, k], 2*(n/k) - 1, 1]; Table[T[n-k+1, k], {n, 1, 10}, {k,1, n}] //Flatten (* Amiram Eldar, Jun 29 2020 *)

T(n,k) = A171233(n,k) - A077049(n,k).

More terms from Amiram Eldar, Jun 29 2020

cp's OEIS Frontend

A077052 Right Moebius transformation matrix, M, by antidiagonals.

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A171233 Array, T(n,k) = 2*(n/k), if n mod k = 0; otherwise, T(n,k) = 1. Read by antidiagonals.

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A247146 As a binary numeral, the bit 2^(m-1) of a(n) is 1 iff m is a proper divisor of n.

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A171232 Array read by antidiagonals, T(n,k) = 2*(n/k) - 1, if n mod k = 0; otherwise, T(n,k) = 1.

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