A061502 -id:A061502 - cp's OEIS frontend

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

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.

A356535 a(n) = Sum_{k=1..n} sigma_2(k)^2.

Original entry on oeis.org

1, 26, 126, 567, 1243, 3743, 6243, 13468, 21749, 38649, 53533, 97633, 126533, 189033, 256633, 372914, 457014, 664039, 795083, 1093199, 1343199, 1715299, 1996199, 2718699, 3142500, 3865000, 4537400, 5639900, 6348864, 8038864, 8964308, 10827533, 12315933, 14418433

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356533.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001157, A057434, A061502, A072379, A356536.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[2, k]^2, {k, 1, n}], {n, 1, 40}]

a(n) = sum(k=1, n, sigma(k, 2)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ 189 * zeta(3)^2 * zeta(5) * n^5 / Pi^6.

A356536 a(n) = Sum_{k=1..n} sigma_3(k)^2.

Original entry on oeis.org

1, 82, 866, 6195, 22071, 85575, 203911, 546136, 1119185, 2405141, 4179365, 8357301, 13188505, 22773721, 35220505, 57132266, 81279662, 127696631, 174756231, 259359435, 352134859, 495847003, 643907227, 912211627, 1160305628, 1551633152, 1969426752, 2600039296

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356534.

In general, for m>0, Sum_{k=1..n} sigma_m(k)^2 ~ zeta(2*m+1) * zeta(m+1)^2 * n^(2*m+1) / ((2*m+1) * zeta(2*m+2)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001158, A057434, A061502, A072379, A356535.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[3, k]^2, {k, 1, n}], {n, 1, 40}]
Accumulate[DivisorSigma[3,Range[40]]^2] (* This program is much more efficient than the first program above. *) (* Harvey P. Dale, Feb 27 2023 *)

a(n) = sum(k=1, n, sigma(k, 3)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ zeta(7) * n^7 / 6.

A319089 a(n) = tau(n)^3, where tau is A000005.

Original entry on oeis.org

1, 8, 8, 27, 8, 64, 8, 64, 27, 64, 8, 216, 8, 64, 64, 125, 8, 216, 8, 216, 64, 64, 8, 512, 27, 64, 64, 216, 8, 512, 8, 216, 64, 64, 64, 729, 8, 64, 64, 512, 8, 512, 8, 216, 216, 64, 8, 1000, 27, 216, 64, 216, 8, 512, 64, 512, 64, 64, 8, 1728, 8, 64, 216, 343

Offset: 1

Vaclav Kotesovec, Sep 10 2018

Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A006218, A035116, A061502, A318755 (partial sums).

with(numtheory): seq(tau(n)^3, n=1..100); # Ridouane Oudra, Mar 07 2023

Mathematica
```
DivisorSigma[0, Range[100]]^3
```

a(n) = numdiv(n)^3; \\ Altug Alkan, Sep 10 2018

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X + X^2)/(1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Mar 09 2023

Multiplicative with a(p^e) = (e+1)^3. - Amiram Eldar, Dec 31 2022
a(n) = Sum_{d1|n} Sum_{d2|n} tau(d1*d2). - Ridouane Oudra, Mar 07 2023
From Vaclav Kotesovec, Mar 09 2023: (Start)
Dirichlet g.f.: Product_{p prime} p^(2*s) * (1 + 4*p^s + p^(2*s)) / (p^s - 1)^4.
Dirichlet g.f.: zeta(s)^8 * Product_{p prime} (1 - 9/p^(2*s) + 16/p^(3*s) - 9/p^(4*s) + 1/p^(6*s)), (with a product that converges for s=1). (End)

A268756 Number of triples x, y, r such that r divides xy, r divides at least one of x or y, and xy <= n.

Original entry on oeis.org

1, 5, 9, 17, 21, 35, 39, 53, 61, 75, 79, 107, 111, 125, 139, 160, 164, 192, 196, 224, 238, 252, 256, 304, 312, 326, 340, 368, 372, 418, 422, 452, 466, 480, 494, 550, 554, 568, 582, 630, 634, 680, 684, 712, 740, 754, 758, 830, 838, 866, 880, 908, 912, 960, 974

Offset: 1

Michel Marcus, Feb 13 2016

nonn

a(n) - a(n - 1) only depends on the prime signature of n. - David A. Corneth, Aug 30 2018

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI prog
Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, arXiv:1602.03555 [math.HO], 2016. See A(n) p. 2.
Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, The American Mathematical Monthly, Vol. 123, No. 9 (2016), 924-927.

Cf. A061502, A268732.

a(n) = {s = 0; for (x=1, n, for (y = 1, n, if (x*y <= n, s += sum(r = 1, x*y, !(x*y % r) && (!(x % r) || !(y % r)));););); s;}

\\ See PARI link \\ David A. Corneth, Aug 30 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A356535 a(n) = Sum_{k=1..n} sigma_2(k)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A356536 a(n) = Sum_{k=1..n} sigma_3(k)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A319089 a(n) = tau(n)^3, where tau is A000005.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A268756 Number of triples x, y, r such that r divides x*y, r divides at least one of x or y, and x*y <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A268756 Number of triples x, y, r such that r divides xy, r divides at least one of x or y, and xy <= n.