A143127 -id:A143127 - cp's OEIS frontend

A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

1, 9, 21, 57, 77, 173, 201, 329, 410, 570, 614, 1046, 1098, 1322, 1562, 1962, 2030, 2678, 2754, 3474, 3810, 4162, 4254, 5790, 6015, 6431, 6863, 7871, 7987, 9907, 10031, 11183, 11711, 12255, 12815, 15731, 15879, 16487, 17111, 19671, 19835, 22523, 22695, 24279

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320897.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^2 * (3*(Pi^6*(-1 - 24*g^2 + 32*g^3 + g*(8 - 96*s1) + 16*s1 + 16*s2) - 13824*z1^3 + 576*Pi^2*z1*((-1 + 8*g)*z1 + 4*z2) - 8*Pi^4*(3*(1 - 8*g + 24*g^2 - 16*s1)*z1 - 6*z2 + 48*g*z2 + 8*z3)) + 6*(Pi^6*(1 - 8*g + 24*g^2 - 16*s1) + 576*Pi^2*z1^2 - 24*Pi^4*(-z1 + 8*g*z1 + 2*z2))*log(n) + 6*((-1 + 8*g)*Pi^6 - 24*Pi^4*z1)*log(n)^2 + 4*Pi^6*log(n)^3) / (8*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

Original entry on oeis.org

1, 17, 53, 197, 297, 873, 1069, 2093, 2822, 4422, 4906, 10090, 10766, 13902, 17502, 23902, 25058, 36722, 38166, 52566, 59622, 67366, 69482, 106346, 111971, 122787, 134451, 162675, 166039, 223639, 227483, 264347, 281771, 300267, 319867, 424843, 430319, 453423

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

In general, for m>=0, Sum_{k=1..n} k^m * tau(k)^2 ~ n^(m+1) * (log(n))^3 / ((m+1) * Pi^2).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320896.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k^2*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k^2*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^3 * (2*Pi^6*(-1 + 12*g - 54*g^2 + 108*g^3 + 36*s1 - 324*g*s1 + 54*s2) - 93312*z1^3 + 2592*Pi^2*z1*(-z1 + 12*g*z1 + 6*z2) - 72*Pi^4*(z1 - 12*g*z1 + 54*g^2*z1 - 36*s1*z1 - 3*z2 + 36*g*z2 + 6*z3) + 6*(Pi^6*(1 - 12*g + 54*g^2 - 36*s1) + 1296*Pi^2*z1^2 - 36*Pi^4*(-z1 + 12*g*z1 + 3*z2))*log(n) + 9*((-1 + 12*g)*Pi^6 - 36*Pi^4*z1)*log(n)^2 + 9*Pi^6*log(n)^3) / (27*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

A348392 Row sums of the irregular triangle A348389.

Original entry on oeis.org

2, 5, 13, 18, 36, 43, 67, 85, 115, 126, 186, 199, 241, 286, 350, 367, 457, 476, 576, 639, 705, 728, 896, 946, 1024, 1105, 1245, 1274, 1484, 1515, 1675, 1774, 1876, 1981, 2269, 2306, 2420, 2537, 2817

Offset: 2

Wolfdieter Lang, Oct 31 2021

			n = 4: Compare the row of an array with all multiples of k <= n, for k = 1,2, ..., n with the row of A348389:
All multiples of k <= 4 for k = 1..4:  [1 2 3 4|2 4|3|4] with row sum A143127(4) = 23.
A348389 row 4: [2 3 4|4] with 1, 2, 3 and 4 missing: row sum is 23 - 4*5/2  = 13. Hence a(4) = A143127(4) - A000217(4).
Also: a(4) =  A348391(4) - A153485(4) = 18 - 5 = 13.

Cf. A000217, A002541, A143127, A348389.

a(n) = Sum_{m=1.. A002541(n)} A348389(n, m), for n >= 2.
a(n) = A143127(n) - A000217(n).
a(n) = A348391(n) - A153485(n).

A350108 a(n) = Sum_{k=1..n} k * floor(n/k)^3.

Original entry on oeis.org

1, 10, 32, 87, 153, 309, 443, 722, 1005, 1443, 1785, 2605, 3087, 3951, 4875, 6154, 6988, 8809, 9855, 12057, 13853, 16001, 17543, 21347, 23478, 26484, 29440, 33696, 36162, 41994, 44816, 50351, 54755, 59909, 64577, 73524, 77558, 84002, 90142, 100072, 105034

Offset: 1

Seiichi Manyama, Dec 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column 3 of A350106.

Cf. A024916, A143128, A143127, A318742.

a[n_] := Sum[k * Floor[n/k]^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[(1 + 3*k)*DivisorSigma[1, k] - 3*k*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)

PARI
```
a(n) = sum(k=1, n, k*(n\k)^3);
```

a(n) = sum(k=1, n, k*sumdiv(k, d, (d^3-(d-1)^3)/d));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1-x^k)^2)/(1-x))

from math import isqrt
def A350108(n): return -(s:=isqrt(n))**4*(s+1)+sum((q:=n//k)*(k**2*(3*(q+1))+k*(q*((q<<1)-3)-3)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 31 2023

a(n) = Sum_{k=1..n} k * Sum_{d|k} (d^3 - (d - 1)^3)/d.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 - x^k)^2.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A024916(n) + 3*A143128(n) - 3*A143127(n).
a(n) ~ Pi^2*n^3/6 - 3*n^2*log(n)/2. (End)

A110661 Triangle read by rows: T(n,k) = total number of divisors of k, k+1, ..., n (1 <= k <= n).

Original entry on oeis.org

1, 3, 2, 5, 4, 2, 8, 7, 5, 3, 10, 9, 7, 5, 2, 14, 13, 11, 9, 6, 4, 16, 15, 13, 11, 8, 6, 2, 20, 19, 17, 15, 12, 10, 6, 4, 23, 22, 20, 18, 15, 13, 9, 7, 3, 27, 26, 24, 22, 19, 17, 13, 11, 7, 4, 29, 28, 26, 24, 21, 19, 15, 13, 9, 6, 2, 35, 34, 32, 30, 27, 25, 21, 19, 15, 12, 8, 6, 37, 36, 34

Offset: 1

Emeric Deutsch, Aug 02 2005

Equals A000012 * (A000005 * 0^(n-k)) * A000012, 1 <= k <= n. - Gary W. Adamson, Jul 26 2008

Row sums = A143127. - Gary W. Adamson, Jul 26 2008

			T(4,2)=7 because 2 has 2 divisors, 3 has 2 divisors and 4 has 3 divisors.
Triangle begins:
   1;
   3, 2;
   5, 4, 2;
   8, 7, 5, 3;
  10, 9, 7, 5, 2;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000005, A006218.

Cf. A143127.

with(numtheory): T:=(n,k)->add(tau(j),j=k..n): for n from 1 to 13 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

T[n_, n_] := DivisorSigma[0, n]; T[n_, k_] := Sum[DivisorSigma[0, j], {j, k, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 03 2017 *)

T(n, k) = Sum_{j=k..n} tau(j), where tau(j) is the number of divisors of j, and 1 <= k <= n.
T(n,n) = tau(n) = A000005(n) = number of divisors of n.
T(n,1) = Sum_{j=1..n} tau(j) = A006218(n).

A143310 Triangle read by rows, A000012 * A127446, 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 6, 2, 3, 10, 6, 3, 4, 15, 6, 3, 4, 5, 21, 12, 9, 4, 5, 6, 28, 12, 9, 4, 5, 6, 7, 36, 20, 9, 12, 5, 6, 7, 8, 45, 20, 18, 12, 5, 6, 7, 8, 9, 55, 30, 18, 12, 15, 6, 7, 8, 9, 10, 66, 30, 18, 12, 15, 6, 7, 8, 9, 10, 11, 78, 42, 30, 24, 15, 18, 7, 8, 9, 10, 11, 12

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = A143127: (1, 5, 11, 23, 33, 57, 71, ...).

			First few rows of the triangle:
   1;
   3,  2;
   6,  2,  3;
  10,  6,  3,  4;
  15,  6,  3,  4,  5;
  21, 12,  9,  4,  5,  6;
  28, 12,  9,  4,  5,  6,  7;
  36, 20,  9, 12,  5,  6,  7,  8;
  ...

Cf. A127446, A143127.

Triangle read by rows, A000012 * A127446, 1 <= k <= n.

Conjecture: T(n,k) = Sum_{i=1..n} [i*k <= n]*i*k. - Mats Granvik, Feb 26 2021

Duplicate a(11)-a(15) terms removed by Mats Granvik, Feb 26 2021

A344746 a(n) = Sum_{k=1..n} d(k) * k^c(n/k), where c(n) = 1 - ceiling(n) + floor(n).

Original entry on oeis.org

1, 5, 9, 19, 18, 40, 28, 59, 51, 73, 49, 136, 61, 107, 113, 164, 84, 210, 96, 235, 166, 180, 120, 397, 167, 217, 227, 338, 159, 469, 173, 419, 275, 293, 287, 682, 214, 332, 330, 667, 240, 666, 254, 549, 538, 412, 280, 1056, 357, 619, 447, 658, 323, 907, 475, 944, 507, 533, 365

Offset: 1

Wesley Ivan Hurt, Jun 02 2021

nonn

For 1 <= k <= n, if k|n then add k * d(k), otherwise add d(k).

If p is prime, a(p) = Sum_{k=1..p} d(k) * k^c(p/k) = 2*p + Sum_{k=1..p-1} d(k) = 2*p - 2 + d(p) + Sum_{k=1..p-1} d(k) = 2*p - 2 + Sum_{k=1..p} d(k).

			a(8) = Sum_{k=1..8} d(k) * k^c(8/k) = d(1)*1^1 + d(2)*2^1 + d(3)*3^0 + d(4)*4^1 + d(5)*5^0 + d(6)*6^0 + d(7)*7^0 + d(8)*8^1 = 1*1 + 2*2 + 2*1 + 3*4 + 2*1 + 4*1 + 2*1 + 4*8 = 59.

Cf. A000005 (tau), A006218, A143127.

Table[Sum[DivisorSigma[0, k] k^(1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

cp's OEIS Frontend

A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

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A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

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A348392 Row sums of the irregular triangle A348389.

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A350108 a(n) = Sum_{k=1..n} k * floor(n/k)^3.

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A110661 Triangle read by rows: T(n,k) = total number of divisors of k, k+1, ..., n (1 <= k <= n).

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A143310 Triangle read by rows, A000012 * A127446, 1 <= k <= n.

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A344746 a(n) = Sum_{k=1..n} d(k) * k^c(n/k), where c(n) = 1 - ceiling(n) + floor(n).

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