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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
OFFSET
1,2
COMMENTS
Number of divisors of n plus the sum of all the remainders modulo the numbers below n.
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
Sequence in context: A099523 A049806 A158381 * A341270 A091515 A036405
KEYWORD
nonn
AUTHOR
EXTENSIONS
169 inserted by R. J. Mathar, Jul 27 2009
STATUS
approved