A356534 -id:A356534 - cp's OEIS frontend

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

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.

A379812 a(n) = sigma_1(n) * sigma_2(n).

Original entry on oeis.org

1, 15, 40, 147, 156, 600, 400, 1275, 1183, 2340, 1464, 5880, 2380, 6000, 6240, 10571, 5220, 17745, 7240, 22932, 16000, 21960, 12720, 51000, 20181, 35700, 32800, 58800, 25260, 93600, 30784, 85995, 58560, 78300, 62400, 173901, 52060, 108600, 95200, 198900, 70644

Offset: 1

Amiram Eldar, Jan 03 2025

Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. Balasubramanian and K. Ramachandra, The place of an identity of Ramanujan in prime number theory, Proceedings of the Indian Academy of Sciences, Section A, Vol. 83. No. 4 (1976), pp. 156-165.
Yoichi Motohashi, On an identity of Ramanujan, Hardy-Ramanujan Journal, Vol. 41 (2019), pp. 48-49.
Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.
Bertram Martin Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, Vol. s2-21, No. 1 (1923), pp. 235-255.

Cf. A000203, A001157, A072861, A086666, A092346, A158274, A158275, A356533, A356534, A379813, A379814.

a[n_] := Times @@ DivisorSigma[{1, 2}, n]; Array[a, 50]

a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 2);}

a(n) = A000203(n) * A001157(n).
Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(2*e+2)-1) / ((p-1) * (p^2-1)).
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) / zeta(2*s-3).
In general, Dirichlet g.f. of  sigma_i(n) * sigma_j(n): zeta(s) * zeta(s-i) * zeta(s-j) * zeta(s-i-j) / zeta(2*s-i-j) (Ramanujan, 1916).
Sum_{k=1..n}  a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(3) * zeta(4) / zeta(5) = zeta(3) * Pi^6 / (540*zeta(5)) = 2.06386841111121962734... .
In general, Sum_{k=1..n}  sigma_i(k) * sigma_j(k) ~ c(i,j) * n^(i+j+1) / (i+j+1), for i, j >= 1, where c(i,j) = zeta(i+1) * zeta(j+1) * zeta(i+j+1) / zeta(i+j+2).
G.f.: Sum_{k>=1} Sum_{l>=1} k*l^2*x^lcm(k, l)/(1 - x^lcm(k, l)). - Miles Wilson, Jul 10 2025

A379813 a(n) = sigma_1(n) * sigma_3(n).

Original entry on oeis.org

1, 27, 112, 511, 756, 3024, 2752, 8775, 9841, 20412, 15984, 57232, 30772, 74304, 84672, 145111, 88452, 265707, 137200, 386316, 308224, 431568, 292032, 982800, 488281, 830844, 817600, 1406272, 731700, 2286144, 953344, 2359287, 1790208, 2388204, 2080512, 5028751

Offset: 1

Amiram Eldar, Jan 03 2025

See A379812 for more links and Ramanujan's general formula.

Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.

Cf. A000203, A001158, A013663, A072861, A091258, A091259, A092348, A092345, A356533, A356534, A379812, A379814.

a[n_] := Times @@ DivisorSigma[{1, 3}, n]; Array[a, 50]

a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 3);}

a(n) = A000203(n) * A001158(n).
Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(3*e+3)-1) / ((p-1) * (p^3-1)).
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-3) * zeta(s-4) / zeta(2*s-4).
Sum_{k=1..n}  a(k) ~ c * n^5 / 5, where c = 7 * zeta(5) / 4 = 1.81462357150089737107... .

cp's OEIS Frontend

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

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

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A379812 a(n) = sigma_1(n) * sigma_2(n).

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A379813 a(n) = sigma_1(n) * sigma_3(n).

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