A079247 Number of pairs (p,q), 0 <= p < q, such that p+q divides n.

1, 2, 3, 4, 4, 7, 5, 8, 8, 10, 7, 15, 8, 13, 14, 16, 10, 21, 11, 22, 18, 19, 13, 31, 17, 22, 22, 29, 16, 38, 17, 32, 26, 28, 26, 47, 20, 31, 30, 46, 22, 50, 23, 43, 42, 37, 25, 63, 30, 48, 38, 50, 28, 62, 38, 61, 42, 46, 31, 86, 32, 49, 55, 64, 44, 74, 35, 64, 50, 74, 37, 99, 38

Offset: 1

Author

Vladeta Jovovic, Feb 03 2003

Keywords

• easy
• nonn

Comments

Equals left border of triangle A158951. - Gary W. Adamson, Mar 31 2009

Equals row sums of triangle A168509. - Gary W. Adamson, Nov 27 2009

Let c(d_x(n)) = (d_x(n) + 1) / 2 if d_x(n) == 1 (mod 2), and d_x(n) / 2 if d_x(n) == 0 (mod 2), where d_x(n) is the x-th divisor of n, 1 <= d_x(n) <= n, and c(d_x(n)) denotes the cardinality of said divisor within the ordered set of naturals sharing its parity. Then, a(n) = Sum_{i=1..A000005(n)} c(d_i(n)). - Christopher Hohl, Apr 16 2019

Examples

			There are 7 pairs (p,q), 0 <= p < q, such that p+q divides 6: (0,1), (0,2), (0,3), (0,6), (1, 2), (1, 5), (2, 4); thus a(6) = 7.
G.f. = x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 5*x^7 + 8*x^8 + 8*x^9 + ...
		

Crossrefs

Cf. A069734, A008619, A158951, A168509, A001227.

Cf. A000203, A086670.

Programs

Maple

with(numtheory): seq((sigma(n)+tau(2*n)-tau(n))/2,n=1 .. 80); # - Ridouane Oudra, Sep 06 2020

PARI

{a(n) = if( n<1, 0, sumdiv( n, d, (1 + d)\2))} /* Michael Somos, Jun 11 2003 */

Formula

Inverse Moebius transform of A008619 (offset 1). - Michael Somos, Jun 11 2003

G.f.: Sum_{k>=1} x^k / ((1 - x^k) * (1 - x^(2*k))). - Michael Somos, Jun 11 2003

G.f.: Sum_{n>=1} A110654(n)*x^n/(1-x^n). - Mircea Merca, Feb 26 2014

a(n) = (1/2)*(A000203(n) + A001227(n)). - Ridouane Oudra, Sep 06 2020

a(n) = A000203(n) - A086670(n). - Ridouane Oudra, Nov 25 2022