A130212 -id:A130212 - cp's OEIS frontend

A130211 Triangle read by rows: matrix product A054522 * A000012.

1, 2, 1, 3, 2, 2, 4, 3, 2, 2, 5, 4, 4, 4, 4, 6, 5, 4, 2, 2, 2, 7, 6, 6, 6, 6, 6, 6, 8, 7, 6, 6, 4, 4, 4, 4, 9, 8, 8, 6, 6, 6, 6, 6, 6, 10, 9, 8, 8, 8, 4, 4, 4, 4, 4, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 11, 10, 8, 6, 6, 4, 4, 4, 4, 4, 4, 13, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Gary W. Adamson, May 17 2007

			First few rows of the triangle are:
1;
2, 1;
3, 2, 2;
4, 3, 2, 2;
5, 4, 4, 4, 4;
6, 5, 4, 2, 2, 2;
7, 6, 6, 6, 6, 6, 6;
8, 7, 6, 6, 4, 4, 4, 4;
...

Cf. A000010, A054522, A130212 (product with swapped order), A057660 (row sums).

A130211 := proc(n,k)
    add( A054522(n,j),j=k..n) ;
end proc:
seq(seq(A130211(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 06 2016

A054522 * A000012 as infinite lower triangular matrices.

T(n,n) = A000010(n).

A181550 T(n,k) = floor(n/k)*A181549(k), triangle read by rows.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 6, 4, 5, 5, 6, 4, 5, 6, 6, 9, 8, 5, 6, 12, 7, 9, 8, 5, 6, 12, 8, 8, 12, 8, 10, 6, 12, 8, 10, 9, 12, 12, 10, 6, 12, 8, 10, 11, 10, 15, 12, 10, 12, 12, 8, 10, 11, 18, 11, 15, 12, 10, 12, 12, 8, 10, 11, 18, 12, 12, 18, 16, 15, 12, 24, 8, 10, 11, 18, 12, 20

Offset: 1

Peter Luschny, Oct 30 2010

A181549(n) = sum{k|n} k mu_2(n/k), a variant of Euler's phi function relative to the Moebius function of order 2.

			1
2, 3
3, 3, 4
4, 6, 4, 5
5, 6, 4, 5, 6
6, 9, 8, 5, 6, 12
7, 9, 8, 5, 6, 12, 8
8, 12, 8, 10, 6, 12, 8, 10

Peter Luschny, Sequences related to Euler's totient function.

Cf. A130212, A181549.

A181550 := (n,k) -> iquo(n,k)*A181549(k);

mu2[1] = 1; mu2[n_] := Sum[Boole[Divisible[n, d^2]]*MoebiusMu[n/d^2]*MoebiusMu[n/d], {d, Divisors[n]}]; A181549[n_] := Sum[k*mu2[n/k], {k, Divisors[n]}]; t[n_, k_] := Floor[n/k]*A181549[k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 05 2014 *)

A181552 T(n,k) = gcd(n,k) A181549(k), triangle read by rows.

Original entry on oeis.org

1, 1, 6, 1, 3, 12, 1, 6, 4, 20, 1, 3, 4, 5, 30, 1, 6, 12, 10, 6, 72, 1, 3, 4, 5, 6, 12, 56, 1, 6, 4, 20, 6, 24, 8, 80, 1, 3, 12, 5, 6, 36, 8, 10, 99, 1, 6, 4, 10, 30, 24, 8, 20, 11, 180, 1, 3, 4, 5, 6, 12, 8, 10, 11, 18, 132, 1, 6, 12, 20, 6, 72, 8, 40, 33, 36, 12, 240

Offset: 1

Peter Luschny, Oct 30 2010

A181549(n) = sum{k|n} k mu_2(n/k) is a variant of Euler's phi function relative to the Moebius function of order 2.

			1,
1,6,
1,3,12,
1,6,.4,20,
1,3,.4,.5,30,
1,6,12,10,.6,72,
1,3,.4,.5,.6,12,56,
1,6,.4,20,.6,24,.8,80,

Peter Luschny, Sequences related to Euler's totient function.

Cf. A130212, A181538, row sums of triangle is A181553.

A181552 := (n,k) -> igcd(n,k)*A181549(k);

mu2[1] = 1; mu2[n_] := Sum[Boole[Divisible[n, d^2]]*MoebiusMu[n/d^2]*MoebiusMu[n/d], {d, Divisors[n]}]; A181549[n_] := Sum[k*mu2[n/k], {k, Divisors[n]}]; t[n_, k_] := GCD[n, k]*A181549[k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 05 2014 *)

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A130211 Triangle read by rows: matrix product A054522 * A000012.

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A181550 T(n,k) = floor(n/k)*A181549(k), triangle read by rows.

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A181552 T(n,k) = gcd(n,k) A181549(k), triangle read by rows.

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Programs

Maple

Mathematica