A054521 -id:A054521 - cp's OEIS frontend

A143615 Triangle A054521 * A000005 as a vector.

1, 1, 3, 3, 8, 3, 14, 7, 14, 8, 27, 7, 35, 12, 20, 18, 50, 11, 58, 16, 35, 24, 74, 15, 68, 29, 54, 29, 101, 15, 111, 39, 64, 41, 84, 26, 140, 47, 78, 40, 158, 24, 168, 51, 75, 61, 186, 34, 170, 49, 111, 66, 217, 39, 160, 65, 131, 80, 247, 32, 261, 84, 122, 92, 197, 45, 292, 92, 162, 60, 312, 55, 326, 104, 135, 106, 263, 55, 356, 85, 206

Offset: 1

Gary W. Adamson, Aug 27 2008

nonn

			a(8) = 7 since the relative primes of 8 are (1, 3, 5, 7). a(8) = d(1) + d(3) + d(5) + d(7) = 1 + 2 + 2 + 2. Or, a(8) = 7 = (1, 0, 1, 0, 1, 0, 1, 0) dot (1, 2, 2, 3, 2, 4, 2, 4), where (1, 0, 1, 0, 1, 0, 1, 0) = row 8 of triangle A054521 and d(n) = (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2,...).
a(7) = 14 = (1, 1, 1, 1, 1, 1, 0) dot (1, 2, 2, 3, 2, 4, 2) = (d(1) + d(2) + d(3) + d(4) + d(5) + d(6)).

Cf. A054521, A000005, A143614.

A143615 := proc(n)
    local a,m;
    a := 0 ;
    for m from 1 to n do
        if gcd(m,n) = 1 then
        a := a+numtheory[tau](m) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 08 2012

Triangle A054521 * A000005 as a vector; where 1's indicate the relative primes of n by rows and A000005 = d(n): (1, 2, 2, 3, 2, 4, 2, 4, 3,...)

a(n) = Sum_{ m=1..n and gcd(n,m)=1 } tau(m), where tau(m)=A000005(m). A211932(n)+a(n) = A006218(n). - Naohiro Nomoto, Aug 05 2012

More terms from Naohiro Nomoto, Aug 05 2012

A143620 Triangle read by rows, square of A054521, 1<=k<=n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 6, 2, 2, 1, 1, 0, 0, 4, 3, 2, 2, 1, 1, 0, 0, 6, 2, 4, 2, 2, 1, 1, 0, 0, 4, 3, 1, 2, 2, 1, 1, 1, 0, 0, 10, 4, 5, 3, 4, 1, 3, 1, 1, 0, 0, 4, 3, 3, 3, 2, 2, 1, 1, 1, 1, 0, 0

Offset: 1

Gary W. Adamson, Aug 27 2008

Left border = phi(n), A000010.

Row sums = A053570: (1, 1, 2, 3, 6, 5, 12, 13,...).

			First few rows of the triangle =
1;
1, 0;
2, 0, 0;
2, 1, 0, 0;
4, 1, 1, 0, 0;
2, 1, 1, 1, 0, 0;
6, 2, 2, 1, 1, 0, 0;
4, 3, 2, 2, 1, 1, 0, 0;
6, 2, 4, 2, 2, 1, 1, 0, 0;
...

Cf. A054521, A000010, A053570.

Triangle read by rows, square of A054521, 1<=k<=n

A144379 Triangle read by rows, first n terms of an array formed by A054521 * A054521(transform).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 4, 2, 6, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 2, 1, 3, 2, 4, 3, 6, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 1, 1, 1, 1, 1, 2, 2, 3, 3, 2, 3, 4, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 1, 1, 1, 2, 2, 2, 3, 3, 2, 3, 4, 3, 5

Offset: 1

Gary W. Adamson, Sep 19 2008

Right border = phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10,...).

Row sums = A125728: (1, 2, 4, 5, 10, 7, 18, 16, 23,...) = the number of positive integers less <=k coprime to both k and n.

			A054521 * A054521(transform) =
1, 1, 1, 1, 1, 1, 1,...
1, 1, 1, 1, 1, 1, 1,...
1, 1, 2, 1, 2, 1, 2,...
1, 1, 1, 2, 2, 1, 2,...
1, 1, 2, 2, 4, 1, 4,...
...
Then extract the lower half of the array including the diagonal, A000010, phi(n); getting triangle A144379:
1;
1, 1;
1, 1, 2
1, 1, 1, 2;
1, 1, 2, 2, 4;
1, 1, 1, 1, 1, 2;
1, 1, 2, 2, 4, 2, 6;
1, 1, 1, 2, 2, 2, 3, 4;
1, 1, 2, 1, 3, 2, 4, 3, 6;
1, 1, 1, 2, 2, 1, 2, 3, 2, 4;
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10;
...

A054521, Cf. A000010, A125728

Given A054521 as an infinite lower triangular matrix, perform A054521(transform). Multiply the result by A054521 getting an array, then extract the first n terms of each row to form a new triangle.

A127447 Triangle defined by the matrix product A127446 * A054521, read by rows 1<=k<=n.

Original entry on oeis.org

1, 4, 0, 6, 3, 0, 12, 0, 4, 0, 10, 5, 5, 5, 0, 24, 6, 0, 0, 6, 0, 14, 7, 7, 7, 7, 7, 0, 32, 0, 16, 0, 8, 0, 8, 0, 27, 18, 0, 9, 9, 0, 9, 9, 0, 40, 10, 20, 10, 0, 0, 10, 0, 10, 0, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 0, 72, 12, 12, 0, 24, 0

Offset: 1

Gary W. Adamson, Jan 14 2007

			First few rows of the triangle are:
1;
4, 0;
6, 3, 0;
12, 0, 4, 0;
10, 5, 5, 5, 0;
24, 6, 0, 0, 6, 0;
14, 7, 7, 7, 7, 7, 0;
32, 0, 16, 0, 8 0, 8, 0;
...

Cf. A038040 (column k=1), A000290 (row sums), A127446, A054521.

A054521 := proc(n, k) if igcd(n,k) = 1 then 1; else 0; fi; end:
A127447 := proc(n,k)
        add( A127446(n,j)*A054521(j,k),j=k..n) ;
end proc:
seq(seq(A127447(n,m),m=1..n),n=1..12) ; # R. J. Mathar, Nov 08 2011

T(n,k) = Sum_{j=k..n} A127446(n,j) * A054521(j,k).

A143614 Triangle read by rows: A054521 * A051731 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 2, 0, 1, 0, 4, 2, 1, 1, 0, 2, 0, 0, 0, 1, 0, 6, 3, 2, 1, 1, 1, 0, 4, 0, 1, 0, 1, 0, 1, 0, 6, 3, 0, 2, 1, 0, 1, 1, 0, 4, 0, 2, 0, 0, 0, 1, 0, 1, 0, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 0, 4, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 6, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Gary W. Adamson, Aug 27 2008

Left border = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...).

Row sums = A143615: (1, 1, 3, 3, 8, 3, 14, 7,...).

			First few rows of the triangle =
1;
1, 0;
2, 1, 0;
2, 0, 1, 0;
4, 2, 1, 1, 0;
2, 0, 0, 0, 1, 0;
6, 3, 2, 1, 1, 1, 0;
4, 0, 1, 0, 1, 0, 1, 0;
6, 3, 0, 2, 1, 0, 1, 1, 0;
...

Cf. A000010, A051731, A054521, A143615.

A054521 records the relative primes of n, indicated by a 1's in row n, 0 otherwise. A051731 = the inverse Moebius transform, in which 1's by rows indicate the divisors of n, 0 otherwise.

a(96) and a(101) split by Georg Fischer, May 29 2023

A143655 Triangle read by rows, primes not dividing n; A054521 * (A061397 * 0^(n-k)), 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 2, 3, 0, 5, 0, 0, 0, 0, 3, 0, 5, 0, 7, 0, 0, 2, 0, 0, 5, 0, 7, 0, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 7, 0, 0, 0, 11, 0, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 0, 0, 0, 3, 0, 5, 0, 0, 0, 0, 0, 11, 0

Offset: 1

Gary W. Adamson, Aug 28 2008

Row sums = A066911: (0, 0, 2, 3, 5, 5, 10, 15, 14,....)

			First few rows of the triangle =
0;
0, 0;
0, 2, 0;
0, 0, 3, 0;
0, 2, 3, 0, 0;
0, 0, 0, 0, 5, 0;
0, 2, 3, 0, 5, 0, 0;
0, 0, 3, 0, 5, 0, 7, 0;
...
Row 8 has 3 primes < 8 not dividing 8: (3, 5, 7); where (3 + 5 + 7) = A066911(8).

Cf. A061397, A066911, A054521.

Triangle read by rows, A054521 * (A061397 * 0^(n-k)), 1<=k<=n. T(n,k) = prime if k is prime but does not divide n. A054521 = a triangle with row sums phi(n). A061397 = (0, 2, 3, 0, 5, 0, 7,...)

A157030 Triangle read by rows, A156834 * A054521.

Original entry on oeis.org

1, 2, 0, 2, 1, 0, 4, 0, 1, 0, 2, 1, 1, 1, 0, 8, 3, 0, 0, 1, 0, 2, 1, 1, 1, 1, 1, 0, 10, 0, 5, 0, 1, 0, 1, 0, 8, 7, 0, 1, 1, 0, 1, 1, 0, 12, 5, 6, 5, 0, 0, 10, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 34, 10, 10, 0, 7, 0, 10, 0, 0, 1, 0

Offset: 1

Gary W. Adamson and Mats Granvik, Feb 21 2009

Left border = A157019: (1, 2, 2, 4, 2, 8, 2, 10,...).

			First few rows of the triangle =
1;
2, 0;
2, 1, 0;
4, 0, 1, 0;
2, 1, 1, 1, 0;
8, 3, 0, 0, 1, 0;
2, 1, 1, 1, 1, 1, 0;
10, 0, 5, 0, 1, 0, 1, 0;
8, 7, 0, 1, 1, 0, 1, 1, 0;
12, 5, 6, 5, 0, 0, 10, 1, 0;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
34, 10, 10, 0, 7, 0, 1, 0, 0, 0, 1, 0;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
...

Cf. A156834 (row sums), A054521.

Triangle read by rows, A156834 * A054521; as infinite lower triangular matrices.

A157031 Triangle A054521 * A157019, where A054521 = an infinite lower triangular matrix and A157019 = a vector [1, 2, 2, 4, 2, 8, 2, 10, 8, ...].

Original entry on oeis.org

1, 1, 3, 3, 9, 3, 19, 7, 21, 13, 51, 7, 87, 17, 39, 51, 175, 11, 239, 21, 169, 111, 415, 15, 489, 185, 313, 219, 1017, 15, 1413, 283, 763, 415, 981, 513, 3057, 839, 1259, 497, 4425, 93, 5605, 893, 1311, 2259, 7505, 521, 8267, 1429, 5473, 3311, 13821, 1449, 11135, 4095

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 21 2009

nonn

			a(4) = 3 = (1, 0, 1, 0) dot (1, 2, 2, 4) = (1 + 0 + 2 + 0), where (1, 0, 1, 0) equals row 4 of triangle A054521.

Cf. A157019, A054521.

A157031 := proc(n)
    add(A054521(n,k)*A157019(k),k=1..n) ;
end proc: # R. J. Mathar, Jul 21 2015

A164295 Triangle T(n,k) read by rows: sum of the triangles A054521 and A051731.

Original entry on oeis.org

2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1

Offset: 1

Roger L. Bagula and Mats Granvik, Aug 12 2009

Zeros in the table, for example T(6,4)=0, indicate that the row and column indices n and k are not coprime and in addition that there is a nonzero remainder n (mod k).

			The table starts
2
2, 1
2, 1, 1
2, 1, 1, 1
2, 1, 1, 1, 1
2, 1, 1, 0, 1, 1
2, 1, 1, 1, 1, 1, 1
2, 1, 1, 1, 1, 0, 1, 1
2, 1, 1, 1, 1, 0, 1, 1, 1
2, 1, 1, 0, 1, 0, 1, 0, 1, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A054521, A051731.

A054521 := proc(n,k) if gcd(n,k) = 1 then 1; else 0 ; fi; end:
A051731 := proc(n,k) if (n mod k) = 0 then 1; else 0 ; fi; end:
A164295 := proc(n,k) A054521(n,k)+A051731(n,k) ; end: seq(seq(A164295(n,k),k=1..n),n=1..10) ;

T[n_, k_] = If[Mod[n, k] == 0, 1, 0] + If[GCD[n, k] == 1, 1, 0];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}]; Flatten[%]

T(n,k) = A054521(n,k) + A051731(n,k), 1<=k<=n, 1<=n.

Edited by the Associate Editors of the OEIS, Aug 28 2009

A349298 Positions k in row n of triangles S(n,k) = T(n,k) = 0, where A054521 = S and A349297 = T, or 0 if there are no such k.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 7, 0, 0, 5, 11, 0, 13, 7, 5, 10, 0, 17, 0, 19, 5, 15, 7, 14, 11, 23, 0, 5, 10, 15, 20, 25, 13, 0, 7, 21, 29, 5, 25, 31, 0, 11, 22, 17, 5, 7, 10, 14, 15, 20, 21, 25, 28, 30, 35, 0, 37, 19, 13, 26, 5, 15, 25, 35, 41, 7, 35, 43, 11, 33, 5, 10, 20, 25, 35, 40

Offset: 1

Michael De Vlieger, Nov 13 2021

Row n is a list of k for which A349297 NOR A054521 is true.
Row p > 3 for p prime has the single term p.
Nonzero terms in this sequence are of the form k*m > 1, where 3-smooth k > 1 in A003586 and 5-rough m > 1 in A007310, with m mod 6 = +/- 1.

			Table T(n,k) for 1 <= n <= 16, replacing 0 with "." and 1 with "*", showing terms in row n of this sequence. Rows with no terms are replaced by 0:
1:   .
2:   .  *
3:   .  .  *
4:   .  *  .  *
5:   .  .  .  .  5
6:   .  *  *  *  .  *
7:   .  .  .  .  .  .  7
8:   .  *  .  *  .  *  .  *
9:   .  .  *  .  .  *  .  .  *
10:  .  *  .  *  5  *  .  *  .  *
11:  .  .  .  .  .  .  .  .  .  . 11
12:  .  *  *  *  .  *  .  *  *  *  .  *
13:  .  .  .  .  .  .  .  .  .  .  .  . 13
14:  .  *  .  *  .  *  7  *  .  *  .  *  .  *
15:  .  .  *  .  5  *  .  .  * 10  .  *  .  .  *
16:  .  *  .  *  .  *  .  *  .  *  .  *  .  *  .  *
---------------------------------------------------
n\k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
Hence, row 5 = {5}, row 7 = {7}, row 11 = {11}, row 13 = {13}, row 14 = {7}, row 15 = {5, 10}, and all other rows 1 <= n <= 16 have no terms, thus are assigned 0 by definition.

Michael De Vlieger, Table of n, a(n) for n = 1..10635 (rows 1 <= n <= 600, flattened)
Michael De Vlieger, 1024-pixel bitmap plotting (n, T(n,k)) in black, otherwise white including for rows containing 0.

Cf. A003586, A007310, A047229, A054521.

With[{nn = 45}, Table[If[Length[#] == 0, {0}, #] &@ Select[Array[# Boole[Xor[Or[Mod[#, 2] == Mod[n, 2] == 0, Mod[#, 3] == Mod[n, 3] == 0], GCD[n, #] != 1]] &, n], # > 0 &], {n, nn}]] // Flatten (* Michael De Vlieger, Dec 08 2021 *)
With[{s = Merge[Map[#1 -> #2 & @@ # &, Position[ImageData[#], 0.]], Identity]}, Array[If[KeyExistsQ[s, #], Lookup[s, #], {0}] &, ImageDimensions[#][[-1]]] // Flatten] &@ Import["https://oeis.org/A349298/a349298.png"] (* Generate 1024 rows stored in the bitmap image, Michael De Vlieger, Dec 08 2021 *)

cp's OEIS Frontend

A143615 Triangle A054521 * A000005 as a vector.

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A143620 Triangle read by rows, square of A054521, 1<=k<=n.

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A144379 Triangle read by rows, first n terms of an array formed by A054521 * A054521(transform).

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A127447 Triangle defined by the matrix product A127446 * A054521, read by rows 1<=k<=n.

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A143614 Triangle read by rows: A054521 * A051731 as infinite lower triangular matrices.

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A143655 Triangle read by rows, primes not dividing n; A054521 * (A061397 * 0^(n-k)), 1<=k<=n.

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A157030 Triangle read by rows, A156834 * A054521.

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A157031 Triangle A054521 * A157019, where A054521 = an infinite lower triangular matrix and A157019 = a vector [1, 2, 2, 4, 2, 8, 2, 10, 8, ...].

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A164295 Triangle T(n,k) read by rows: sum of the triangles A054521 and A051731.

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