A060872 Sum of d*d' over all unordered pairs (d,d') with d*d' = n.

1, 2, 3, 8, 5, 12, 7, 16, 18, 20, 11, 36, 13, 28, 30, 48, 17, 54, 19, 60, 42, 44, 23, 96, 50, 52, 54, 84, 29, 120, 31, 96, 66, 68, 70, 180, 37, 76, 78, 160, 41, 168, 43, 132, 135, 92, 47, 240, 98, 150, 102, 156, 53, 216, 110, 224, 114, 116, 59, 360, 61, 124, 189, 256

Offset: 1

Author

N. J. A. Sloane, May 04 2001

Keywords

• nonn
• easy

Comments

a(n) is also the sum of all parts of all partitions of n into consecutive parts that differ by 2. - Omar E. Pol, May 05 2020

Examples

			a(4)=8 because pairs of factors are 1*4 and 2*2 and 1*4 + 2*2 = 8.
For n = 16 there are three partitions of 16 into consecutive parts that differ by 2 (including 16 as a valid partition). They are [16], [9, 7] and [7, 5, 3, 1]. The sum of the parts is [16] + [9 + 7] + [7 + 5 + 3 + 1] = 48, so a(16) = 48. - _Omar E. Pol_, May 05 2020
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A038548, A060866.

First differences of A083356.

Programs

Magma

[n*Ceiling(DivisorSigma(0, n)/2): n in [1..70]]; // Vincenzo Librandi, Apr 12 2017

Mathematica

Table[ n * Ceiling[ DivisorSigma[0, n] /2 ], {n, 1, 73} ]

PARI

a(n) = n*ceil(numdiv(n)/2); \\ Michel Marcus, Jul 12 2023

Python

from sympy import divisor_count
def A060872(n): return n*(divisor_count(n)+1>>1) # Chai Wah Wu, Jul 11 2023

Formula

a(n) = n * ceiling( d(n)/2) where d is the number of divisors function.

G.f.: x*f'(x), where f(x) = Sum_{k>=1} x^k^2/(1 - x^k). - Ilya Gutkovskiy, Apr 10 2017

a(n) = n*A038548(n). - Omar E. Pol, May 05 2020

Dirichlet g.f.: (zeta(2*s-2) + zeta(s-1)^2)/2. - Vaclav Kotesovec, Oct 21 2024

Extensions

More terms from Robert G. Wilson v, Jun 23 2001