A097988 -id:A097988 - cp's OEIS frontend

A330570 Partial sums of A097988 (d_3(n)^2).

1, 10, 19, 55, 64, 145, 154, 254, 290, 371, 380, 704, 713, 794, 875, 1100, 1109, 1433, 1442, 1766, 1847, 1928, 1937, 2837, 2873, 2954, 3054, 3378, 3387, 4116, 4125, 4566, 4647, 4728, 4809, 6105, 6114, 6195, 6276, 7176, 7185, 7914, 7923, 8247, 8571, 8652, 8661, 10686, 10722, 11046, 11127, 11451, 11460, 12360

Offset: 1

N. J. A. Sloane, Jan 08 2020

nonn

This and the following sequences (and continuing in A331071) were inspired by the papers of Hooley, Indlekofer, Motohashi, Redmond, Titchmarsh, etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Vincenzo Librandi)
V. C. Harris and M. V. Subbarao, On the divisor sum function, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp 399-412; alternative link.
C. Hooley, An Asymptotic Formula in the Theory of Numbers, Proceedings of the London Mathematical Society, Volume s3-7, Issue 1, 1957, Pages 396-413.
Karl-Heinz Indlekofer, Eine asymptotische Formel in der Zahlentheorie (German), Arch. Math. (Basel) 23 (1972), 619-624. MR0318080 (47 #6629).
Yoichi Motohashi, An asymptotic formula in the theory of numbers, Acta Arith. 16 (1969/70), 255-264. MR0266884 (42 #1786).
Don Redmond, An asymptotic formula in the theory of numbers, Math. Ann. 224 (1976), no. 3, 247-268. MR0419386 (54 #7407).
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.

Cf. A007425, A097988.

Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, #]&]^2; Array[a, 60]] (* Vincenzo Librandi, Jan 11 2020 *)

lista(nmax) = {my(s = 0); for(n = 1, nmax, s += vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[,2]))^2; print1(s, ", "));} \\ Amiram Eldar, Apr 19 2024

a(n) ~ c * n * log(n)^8 /8!, where c = Product_{p prime} ((1-1/p)^4 * (1 + 4/p + 1/p^2)) = 0.049321673579400091761... (Titchmarsh, 1942). - Amiram Eldar, Apr 19 2024

A344080 a(n) = Sum_{d|n} tau(d)^n, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 9, 98, 33, 4225, 129, 72354, 20196, 1050625, 2049, 2194099186, 8193, 268468225, 1073807361, 156925970179, 131073, 101629064089930, 524289, 3657261440572306, 4398050705409, 17592194433025, 8388609, 4727105427440383342818, 847322163876, 4503599761588225

Offset: 1

Seiichi Manyama, May 09 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..719

Cf. A007425, A062367, A097988, A279789, A344060, A344081.

a[n_] := DivisorSum[n, DivisorSigma[0, #]^n &]; Array[a, 26] (* Amiram Eldar, May 09 2021 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d)^n);
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (numdiv(k)*x)^k/(1-(numdiv(k)*x)^k)))

G.f.: Sum_{k >= 1} (tau(k) * x)^k/(1 - (tau(k) * x)^k).

If p is prime, a(p) = 1 + 2^p.

A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 9, 86, 33, 4109, 129, 65622, 19692, 1048613, 2049, 2176786526, 8193, 268435589, 1073741865, 152587956247, 131073, 101559956692208, 524289, 3656158441111670, 4398046511241, 17592186046469, 8388609, 4722366482871822065758

Offset: 1

Seiichi Manyama, May 09 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..719

Cf. A007425, A062367, A097988, A279789, A344047, A344080.

a[n_] := DivisorSum[n, DivisorSigma[0, #]^# &]; Array[a, 24] (* Amiram Eldar, May 09 2021 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d)^d);
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (numdiv(k)*x)^k/(1-x^k)))

G.f.: Sum_{k >= 1} (tau(k) * x)^k/(1 - x^k).

If p is prime, a(p) = 1 + 2^p.

A344082 a(n) = n * Sum_{d|n} tau(d)^3 / d, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 10, 11, 47, 13, 110, 15, 158, 60, 130, 19, 517, 21, 150, 143, 441, 25, 600, 27, 611, 165, 190, 31, 1738, 92, 210, 244, 705, 37, 1430, 39, 1098, 209, 250, 195, 2820, 45, 270, 231, 2054, 49, 1650, 51, 893, 780, 310, 55, 4851, 132, 920, 275, 987, 61, 2440, 247, 2370, 297, 370, 67, 6721, 69

Offset: 1

Seiichi Manyama, May 09 2021

László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

Cf. A007429, A013661, A062369, A097988, A344043.

a[n_] := n * DivisorSum[n, DivisorSigma[0, #]^3/# &]; Array[a, 61] (* Amiram Eldar, May 09 2021 *)

PARI
```
a(n) = n*sumdiv(n, d, numdiv(d)^3/d);
```

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

G.f.: Sum_{k >= 1} tau(k)^3 * x^k/(1 - x^k)^2.
If p is prime, a(p) = 8 + p.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)^4 * Product_{p prime} (1 + 4/p^2 + 1/p^4) = 31.237542262502... . - Amiram Eldar, Dec 22 2023
From Peter Bala, Jan 25 2024: (Start)
a(n) = Sum_{d|n, e|n} gcd(d, e) * tau(n/d) * tau(n/e) (the sum is a multiplicative function of n - see Tóth).
Multiplicative: a(p^k) = ( p^(k+2)*(p^2 + 4*p + 1) - p^3*(k + 2)^3 + p^2*(3*k^3 + 15*k^2 + 21*k + 5) - p*(3*k^3 + 12*k^2 + 12*k + 4) + (k + 1)^3 ) / (p - 1)^4. (End)

cp's OEIS Frontend

A330570 Partial sums of A097988 (d_3(n)^2).

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A344080 a(n) = Sum_{d|n} tau(d)^n, where tau(n) is the number of divisors of n.

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A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

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Mathematica

PARI

PARI

Formula

A344082 a(n) = n * Sum_{d|n} tau(d)^3 / d, where tau(n) is the number of divisors of n.

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Mathematica

PARI

PARI

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