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

1,2

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

Links

Table of n, a(n) for n=1..73.

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

Example

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 + ...

Maple

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

Prog

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

Crossrefs

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

Cf. A000203, A086670.

Sequence in context: A053273 A347700 A049988 * A325588 A244903 A342337

Adjacent sequences: A079244 A079245 A079246 * A079248 A079249 A079250

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 03 2003

Status

approved