A079247 - OEIS

%I #33 Dec 30 2022 20:14:30

%S 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,

%T 22,29,16,38,17,32,26,28,26,47,20,31,30,46,22,50,23,43,42,37,25,63,30,

%U 48,38,50,28,62,38,61,42,46,31,86,32,49,55,64,44,74,35,64,50,74,37,99,38

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

%C Equals left border of triangle A158951. - _Gary W. Adamson_, Mar 31 2009

%C Equals row sums of triangle A168509. - _Gary W. Adamson_, Nov 27 2009

%C 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

%F Inverse Moebius transform of A008619 (offset 1). - _Michael Somos_, Jun 11 2003

%F G.f.: Sum_{k>=1} x^k / ((1 - x^k) * (1 - x^(2*k))). - _Michael Somos_, Jun 11 2003

%F G.f.: Sum_{n>=1} A110654(n)*x^n/(1-x^n). - _Mircea Merca_, Feb 26 2014

%F a(n) = (1/2)*(A000203(n) + A001227(n)). - _Ridouane Oudra_, Sep 06 2020

%F a(n) = A000203(n) - A086670(n). - _Ridouane Oudra_, Nov 25 2022

%e 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.

%e 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 + ...

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

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (1 + d)\2))} /* _Michael Somos_, Jun 11 2003 */

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

%Y Cf. A000203, A086670.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Feb 03 2003