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).

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

Author

Gary W. Adamson, Apr 05 2007

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A129234, A129236, A129237.

Cf. A000005, A000203.

Cf. A130307.

Cf. A051731, A143315, A143594.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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

Extensions

Edited by Emeric Deutsch, Apr 17 2007