A163180 - OEIS

%I #11 Oct 23 2023 01:59:38

%S 1,2,3,4,6,7,10,12,15,17,24,23,30,35,40,41,53,53,66,67,74,81,100,93,

%T 106,116,129,130,153,146,169,173,188,201,222,207,235,252,273,266,299,

%U 292,327,334,345,362,405,384,417,426,453,460,507,500,533,528,557,582,637,598,647

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

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

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

%e a(1) = 1 + 0 = 1;

%e a(2) = 2 + 0 = 2;

%e a(3) = 2 + 1 = 3;

%e a(4) = 3 + 1 = 4;

%e a(5) = 2 + 4 = 6.

%p 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

%t Table[DivisorSigma[0,n]+Sum[Mod[n,k],{k,n}],{n,70}] (* _Harvey P. Dale_, Feb 11 2015 *)

%o (Python)

%o from math import isqrt

%o from sympy import divisor_count

%o 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

%Y Cf. A000005, A004125.

%K nonn

%O 1,2

%A _Juri-Stepan Gerasimov_, Jul 22 2009

%E 169 inserted by _R. J. Mathar_, Jul 27 2009