A168011 -id:A168011 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

Omar E. Pol, Nov 16 2009

nonn

			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).

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

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

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

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

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

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

A171762 a(n) = Sum_{k=n^2+1..(n+1)^2-1} tau(k).

Original entry on oeis.org

4, 12, 22, 34, 44, 58, 72, 88, 100, 120, 126, 148, 164, 182, 196, 220, 228, 254, 264, 284, 304, 328, 338, 358, 382, 400, 420, 444, 442, 478, 494, 518, 544, 564, 562, 602, 622, 648, 652, 690, 684, 730, 740, 768, 790, 812, 828, 858, 870, 898, 920, 946, 958, 990

Offset: 1

Giovanni Teofilatto, Dec 18 2009

Cf. A000005, A048691, A168011.

A168011 := proc(n) add( numtheory[tau](k),k=1..n^2+2*n) ; end proc: A048691 := proc(n) numtheory[tau](n^2) ; end proc: A171762 := proc(n) A168011(n)-A168011(n-1)-A048691(n) ; end proc: seq(A171762(n),n=1..80) ; # R. J. Mathar, Jan 25 2010

Array[n \[Function] Sum[DivisorSigma[0, k], {k, n^2 + 1, (n + 1)^2 - 1}], 200] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)

a(n) = A168011(n) - A168011(n-1) - A048691(n). - R. J. Mathar, Jan 25 2010

Definition corrected by Giovanni Teofilatto, Dec 19 2009

More terms from R. J. Mathar and J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

cp's OEIS Frontend

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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A168012 a(n) = sum of all 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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Formula

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A171762 a(n) = Sum_{k=n^2+1..(n+1)^2-1} tau(k).

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