|
|
A163180
|
|
a(n) = tau(n) + Sum_{k=1..n} (n mod k).
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Number of divisors of n plus the sum of all the remainders modulo the numbers below n.
|
|
LINKS
|
|
|
FORMULA
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|