A143315 -id:A143315 - cp's OEIS frontend

A143467 Triangle read by rows, A143315 * A128407, 1<=k<=n.

1, 3, -1, 5, 0, -1, 7, -3, 0, 0, 9, 0, 0, 0, -1, 11, -5, -3, 0, 0, 1, 13, 0, 0, 0, 0, 0, -1, 15, -7, 0, 0, 0, 0, 0, 0, 17, 0, -5, 0, 0, 0, 0, 0, 0, 19, -9, 0, 0, -3, 0, 0, 0, 0, 1, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 23, -11, -7, 0, 0, 3, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Gary W. Adamson, Aug 17 2008

Row sums = A140434: (1, 2, 4, 4, 8, 4, 12, 8, 12,...).

Right border = mu(n), A008683: (1, -1, -1, 0, -1, 1,...).

			First few rows of the triangle =
1;
3, -1;
5, 0, -1;
7, -3, 0, 0;
9, 0, 0, 0, -1;
11, -5, -3, 0, 0, 1;
13, 0, 0, 0, 0, 0, -1;
...

Cf. A008683, A128407, A143315, A143467.

a(66) corrected by Georg Fischer, Aug 14 2023

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

A143469 Triangle read by rows, A000012 * A143315 * A128407, 1<=k<=1.

Original entry on oeis.org

1, 4, -1, 9, -1, -1, 16, -4, -1, 0, 25, -4, -1, 0, -1, 36, -9, -4, 0, -1, 1, 49, -9, -4, 0, -1, 1, -1, 64, -16, -4, 0, -1, 1, -1, 0, 81, -16, -9, 0, -1, 1, -1, 0, 0, 100, -25, -9, 0, -4, 1, -1, 0, 0, 1, 121, -25, -9, 0, -4, 1, -1, 0, 0, 1, -1, 144, -36, -16, 0, -4, 4, -1, 0, 0, 1, -1, 0

Offset: 1

Gary W. Adamson, Aug 17 2008

Row sums = A018805: (1, 3, 7, 11, 19, 23, 35,...).

Right border = mu(n), A008683.

			First few rows of the triangle =
1;
4, -1;
9, -1, -1;
16, -4, -1, 0;
25, -4, -1, 0, -1;
36, -9, -4, 0, -1, 1;
49, -9, -4, 0, -1, 1, -1;
...

Cf. A008683, A018805, A128407, A143315.

a(9) corrected and more terms from Georg Fischer, Aug 14 2023

A129235 a(n) = 2*sigma(n) - tau(n), where tau(n) is the number of divisors of n (A000005) and sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

1, 4, 6, 11, 10, 20, 14, 26, 23, 32, 22, 50, 26, 44, 44, 57, 34, 72, 38, 78, 60, 68, 46, 112, 59, 80, 76, 106, 58, 136, 62, 120, 92, 104, 92, 173, 74, 116, 108, 172, 82, 184, 86, 162, 150, 140, 94, 238, 111, 180, 140, 190, 106, 232, 140, 232, 156, 176, 118, 324, 122, 188

Offset: 1

Gary W. Adamson, Apr 05 2007

nonn

Row sums of A129234. - Emeric Deutsch, Apr 17 2007
Equals row sums of A130307. - Gary W. Adamson, May 20 2007
Equals row sums of triangle A143315. - Gary W. Adamson, Aug 06 2008
Equals A051731 * (1, 3, 5, 7, ...); i.e., the inverse Mobius transform of the odd numbers. Example: a(4) = 11 = (1, 1, 0, 1) * (1, 3, 5, 7) = (1 + 3 + 0 + 7), where (1, 1, 0, 1) = row 4 of A051731. - Gary W. Adamson, Aug 17 2008
Equals row sums of triangle A143594. - Gary W. Adamson, Aug 26 2008

			a(4) = 2*sigma(4) - tau(4) = 2*7 - 3 = 11.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A129234, A129236, A129237.
Cf. A000005, A000203.
Cf. A130307.
Cf. A051731, A143315, A143594.

with(numtheory): seq(2*sigma(n)-tau(n),n=1..75); # Emeric Deutsch, Apr 17 2007
G:=sum(z^k*(k-(k-1)*z^k)/(1-z^k)^2,k=1..100): Gser:=series(G,z=0,80): seq(coeff(Gser,z,n),n=1..75); # Emeric Deutsch, Apr 17 2007

a[n_] := DivisorSum[2n, If[EvenQ[#], #-1, 0]&]; Array[a, 70] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)
Table[2*DivisorSigma[1,n]-DivisorSigma[0,n],{n,80}] (* Harvey P. Dale, Aug 07 2022 *)

a(n)=sumdiv(2*n,d, if(d%2==0, d-1, 0 ) ); /* Joerg Arndt, Oct 07 2012 */

a(n) = 2*sigma(n)-numdiv(n); \\ Altug Alkan, Mar 18 2018

G.f.: Sum_{k>=1} z^k*(k-(k-1)*z^k)/(1-z^k)^2. - Emeric Deutsch, Apr 17 2007
G.f.: Sum_{n>=1} x^n*(1+x^n)/(1-x^n)^2. - Joerg Arndt, May 25 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(2-1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018
a(n) = A222548(n) - A222548(n-1). - Ridouane Oudra, Jul 11 2020

Edited by Emeric Deutsch, Apr 17 2007

cp's OEIS Frontend

A143467 Triangle read by rows, A143315 * A128407, 1<=k<=n.

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

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A143469 Triangle read by rows, A000012 * A143315 * A128407, 1<=k<=1.

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A129235 a(n) = 2*sigma(n) - tau(n), where tau(n) is the number of divisors of n (A000005) and sigma(n) is the sum of divisors of n (A000203).

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