A355750 Sum of the divisors of 2n minus the number of divisors of 2n.

1, 4, 8, 11, 14, 22, 20, 26, 33, 36, 32, 52, 38, 50, 64, 57, 50, 82, 56, 82, 88, 78, 68, 114, 87, 92, 112, 112, 86, 156, 92, 120, 136, 120, 136, 183, 110, 134, 160, 176, 122, 212, 128, 172, 222, 162, 140, 240, 165, 208, 208, 202, 158, 268, 208, 238, 232, 204, 176, 344, 182

Offset: 1

Author

Wesley Ivan Hurt, Jul 15 2022

Keywords

• nonn
• easy

Comments

Consider the partitions of 2n into 2 parts (s,t), where s <= t. a(n) gives the sum of all the quotients t/s such that t/s is an integer. (See example.)

Examples

			a(7) = 20; the partitions of 2*7 = 14 into two parts (s,t) where s <= t are: (1,13), (2,12), (3,11), (4,10), (5,9), (6,8), and (7,7). The sum of the quotients t/s such that each t/s is an integer is then: 13/1 + 12/2 + 7/7 = 13 + 6 + 1 = 20.
		

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A062731, A099777.

Bisection of A065608.

Programs

Mathematica

Table[DivisorSigma[1, 2 n] - DivisorSigma[0, 2 n], {n, 80}]

PARI

a(n) = my(f=factor(2*n)); sigma(f) - numdiv(f); \\ Michel Marcus, Jul 16 2022

Formula

a(n) = sigma(2n) - tau(2n).

a(n) = Sum_{d|2n} (2n-d)/d.

a(n) = A065608(2n) = A000203(2n) - A000005(2n).

a(n) = A062731(n) - A099777(n).

a(n) = Sum_{k=1..n} m*c(m), where m=(2n-k)/k and c(m)=1-ceiling(m)+floor(m).