A168012 -id:A168012 - cp's OEIS frontend

A168011 a(n) = Sum of all numbers of divisors of all numbers < (n+1)^2.

5, 20, 45, 84, 131, 198, 273, 368, 473, 602, 731, 894, 1061, 1252, 1457, 1686, 1917, 2186, 2453, 2752, 3065, 3402, 3743, 4122, 4509, 4918, 5345, 5804, 6249, 6754, 7251, 7780, 8333, 8906, 9477, 10104, 10729, 11386, 12047, 12758, 13445, 14202, 14945

Offset: 1

Omar E. Pol, Nov 16 2009

nonn

Partial sums of A168010.

			a(2)=20 because the numbers < (2+1)^2 are 1,2,3,4,5,6,7 and 8. Then a(2) = d(1)+d(2)+d(3)+d(4)+d(5)+d(6)+d(7)+d(8) = 1+2+2+3+2+4+2+4 = 20, where d(n) is the number of divisor of n (see A000005).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000005, A168010, A168012, A168013.

f[n_] := Plus @@ DivisorSigma[0, Range[(n + 1)^2 - 1]]; Array[f, 43] (* Robert G. Wilson v, Dec 10 2009 *)

def A168011(n):
    m = n*(n+2)
    return (sum(m//k for k in range(1,n+1))<<1)-n**2 # Chai Wah Wu, Oct 23 2023

More terms from Robert G. Wilson v, Dec 10 2009

A168010 a(n) = Sum of all numbers of divisors of all numbers k such that n^2 <= k < (n+1)^2.

Original entry on oeis.org

5, 15, 25, 39, 47, 67, 75, 95, 105, 129, 129, 163, 167, 191, 205, 229, 231, 269, 267, 299, 313, 337, 341, 379, 387, 409, 427, 459, 445, 505, 497, 529, 553, 573, 571, 627, 625, 657, 661, 711, 687, 757, 743, 783, 805, 821, 831, 885, 875, 913, 929, 961, 961, 1011

Offset: 1

Omar E. Pol, Nov 16 2009

nonn

A straightforward approach to calculate a(n) would require computing tau (A000005) for the 2n+1 integers between n^2 and (n+1)^2. Since Sum_{i=1..n} tau(i) can be computed by summing sqrt(n) terms, we can compute a(n) via the summation of n terms of the form 2*(floor(n*(n+2)/i)-floor((n-1)*(n+1)/i)) without the need to compute tau. Similarly for the sequence A168012. - Chai Wah Wu, Oct 24 2023

			a(2) = 15 because the numbers k are 4, 5, 6, 7 and 8 (since 2^2 <= k < 3^2) and d(4) + d(5) + d(6) + d(7) + d(8) = 3 + 2 + 4 + 2 + 4 = 15, where d(n) is the number of divisors of n (see A000005).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from G. C. Greubel)

Cf. A000005, A168011, A168012, A168013.

Table[Total[DivisorSigma[0,Range[n^2,(n+1)^2-1]]],{n,60}] (* Harvey P. Dale, Aug 17 2015 *)

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

def A168010(n):
    a, b = n*(n+2),(n-1)*(n+1)
    return (sum(a//k-b//k for k in range(1,n))<<1)+5 # Chai Wah Wu, Oct 23 2023

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

A168013 a(n) = Sum of all divisors of all numbers < (n+1)^2.

Original entry on oeis.org

8, 56, 189, 491, 1007, 1930, 3276, 5314, 8082, 11973, 16783, 23355, 31314, 41380, 53566, 68510, 85771, 106981, 130973, 159470, 192020, 229762, 271873, 320779, 375031, 436311, 504464, 581422, 664364, 759025, 860907, 973989, 1097783, 1233366, 1378996, 1540522

Offset: 1

Omar E. Pol, Nov 16 2009

nonn

Partial sums of A168012.

			For n=2 the a(2)=56 because the numbers < (2+1)^2 are 1,2,3,4,5,6,7 and 8. Then a(2) = sigma(1)+sigma(2)+sigma(3)+sigma(4)+sigma(5)+sigma(6)+sigma(7)+sigma(8) = 1+3+4+7+6+12+8+15 = 56, where sigma(n) is the sum of divisor of n (see A000203).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000203, A024916, A168010, A168011, A168012.

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

def A168013(n):
    m = n*(n+2)
    return sum((q:=m//k)*((k<<1)+q+1) for k in range(1,n+1))-n**2*(n+1)>>1 # Chai Wah Wu, Oct 23 2023

a(n) = A024916(n^2+2*n). - Jason Yuen, Oct 08 2024

More terms from Sean A. Irvine, Dec 07 2009

cp's OEIS Frontend

A168011 a(n) = Sum of all numbers of divisors of all numbers < (n+1)^2.

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A168010 a(n) = Sum of all numbers of divisors of all numbers k such that n^2 <= k < (n+1)^2.

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A168013 a(n) = Sum of all divisors of all numbers < (n+1)^2.

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