A168012 a(n) = sum of all divisors of all numbers k such that n^2 <= k < (n+1)^2.

8, 48, 133, 302, 516, 923, 1346, 2038, 2768, 3891, 4810, 6572, 7959, 10066, 12186, 14944, 17261, 21210, 23992, 28497, 32550, 37742, 42111, 48906, 54252, 61280, 68153, 76958, 82942, 94661, 101882, 113082, 123794, 135583, 145630, 161526

Offset: 1

Author

Omar E. Pol, Nov 16 2009

Keywords

• nonn

Examples

			a(2) = 48 because the numbers k are 4,5,6,7 and 8 (since 2^2 <= k < 3^2) and sigma(4) + sigma(5) + sigma(6) + sigma(7) + sigma(8) = 7 + 6 + 12 + 8 + 15 = 48, where sigma(n) is the sum of divisors of n (see A000203).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203, A024916, A168010, A168011, A168013.

Programs

Mathematica

A168012[n_]:=Sum[DivisorSigma[1,k],{k,n^2,(n+1)^2-1}];
Array[A168012,50] (* Paolo Xausa, Oct 23 2023 *)

PARI

a(n)=sum(k=n^2,(n+1)^2-1,sigma(k)) \\ Franklin T. Adams-Watters, May 14 2010

Python

def A168012(n):
    a, b = n*(n+2),(n-1)*(n+1)
    return (sum((q:=a//k)*((s:=k<<1)+q+1)-(r:=b//k)*(s+r+1) for k in range(1,n))>>1)+5*n+3 # Chai Wah Wu, Oct 23 2023

Extensions

More terms from Franklin T. Adams-Watters, May 14 2010