A163180 a(n) = tau(n) + Sum_{k=1..n} (n mod k).

1, 2, 3, 4, 6, 7, 10, 12, 15, 17, 24, 23, 30, 35, 40, 41, 53, 53, 66, 67, 74, 81, 100, 93, 106, 116, 129, 130, 153, 146, 169, 173, 188, 201, 222, 207, 235, 252, 273, 266, 299, 292, 327, 334, 345, 362, 405, 384, 417, 426, 453, 460, 507, 500, 533, 528, 557, 582, 637, 598, 647

Offset: 1

Author

Juri-Stepan Gerasimov, Jul 22 2009

Keywords

• nonn

Comments

Number of divisors of n plus the sum of all the remainders modulo the numbers below n.

Examples

			a(1) = 1 + 0 = 1;
a(2) = 2 + 0 = 2;
a(3) = 2 + 1 = 3;
a(4) = 3 + 1 = 4;
a(5) = 2 + 4 = 6.
		

Crossrefs

Cf. A000005, A004125.

Programs

Maple

A004125 := proc(n) add( modp(n,k),k=2..n) ; end: A163180 := proc(n) numtheory[tau](n)+A004125(n) ; end: seq(A163180(n),n=1..80) ; # R. J. Mathar, Jul 27 2009

Mathematica

Table[DivisorSigma[0,n]+Sum[Mod[n,k],{k,n}],{n,70}] (* Harvey P. Dale, Feb 11 2015 *)

Python

from math import isqrt
from sympy import divisor_count
def A163180(n): return divisor_count(n)+n**2+((s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

Formula

a(n) = A000005(n) + A004125(n).

Extensions

169 inserted by R. J. Mathar, Jul 27 2009