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

1,2

Comments

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

Links

Table of n, a(n) for n=1..61.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000005, A004125.

Sequence in context: A099523 A049806 A158381 * A341270 A091515 A036405

Adjacent sequences: A163177 A163178 A163179 * A163181 A163182 A163183

Keyword

nonn

Author

Juri-Stepan Gerasimov, Jul 22 2009

Extensions

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

Status

approved