A013662 -id:A013662 - cp's OEIS frontend

A352031 Sum of the cubes of the odd proper divisors of n.

0, 1, 1, 1, 1, 28, 1, 1, 28, 126, 1, 28, 1, 344, 153, 1, 1, 757, 1, 126, 371, 1332, 1, 28, 126, 2198, 757, 344, 1, 3528, 1, 1, 1359, 4914, 469, 757, 1, 6860, 2225, 126, 1, 9632, 1, 1332, 4257, 12168, 1, 28, 344, 15751, 4941, 2198, 1, 20440, 1457, 344, 6887, 24390, 1, 3528, 1

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 126; a(10) = Sum_{d|10, d<10, d odd} d^3 = 1^3 + 5^3 = 126.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), this sequence (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013662, A051000.

f[2, e_] := 1; f[p_, e_] := (p^(3*e+3) - 1)/(p^3 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^3, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
Table[Total[Select[Most[Divisors[n]],OddQ]^3],{n,70}] (* Harvey P. Dale, Apr 14 2025 *)

a(n) = sumdiv(n/2^valuation(n,2), d, if ((dMichel Marcus, Mar 02 2022

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^3 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A051000(n) - n^3*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)-1)/8 = 0.0102904042... . (End)

A116963 Inverse Moebius transform of the shifted tetrahedral numbers.

Original entry on oeis.org

4, 14, 24, 49, 60, 118, 124, 214, 244, 356, 368, 608, 564, 814, 896, 1183, 1144, 1668, 1544, 2162, 2168, 2678, 2604, 3698, 3336, 4228, 4304, 5344, 4964, 6732, 5988, 7728, 7528, 8924, 8616, 11297, 9884, 12214, 12064, 14668, 13248, 17132, 15184, 18928, 18412, 21038

Offset: 1

Jonathan Vos Post, Mar 31 2006

			a(12) = ((1+1)*(1+2)*(1+3)/6) + ((2+1)*(2+2)*(2+3)/6) + ((3+1)*(3+2)*(3+3)/6) + ((4+1)*(4+2)*(4+3)/6) + ((6+1)*(6+2)*(6+3)/6) + ((12+1)*(12+2)*(12+3)/6) = 4 + 10 + 20 + 35 + 84 + 455 = 608.
a(13) = ((1+1)*(1+2)*(1+3)/6) + ((13+1)*(13+2)*(13+3)/6) = 4 + 560 = 564.

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

See also: A007437 (inverse Moebius transform of triangular numbers).
Cf. A000292, A007437, A007503.
Cf. A013662, A059358, A363604, A363607, A363611.
Cf. A000005, A000203, A001157, A001158.

a[n_] := DivisorSum[n, Binomial[# + 3, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 05 2023 *)

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, 1/(1-x^k)^4-1)) \\ Seiichi Manyama, Jun 12 2023

a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)/6 = Sum_{d|n} A000292(d+1).
G.f.: Sum_{k>0} (1/(1-x^k)^4 - 1). - Seiichi Manyama, Jun 12 2023
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_3(n) + 6*sigma_2(n) + 11*sigma_1(n) + 6*sigma_0(n))/6.
Dirichlet g.f.: zeta(s) * (zeta(s-3) + 6*zeta(s-2) + 11*zeta(s-1) + 6*zeta(s)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 16, 41, 136, 157, 656, 401, 1096, 1121, 2512, 1465, 5576, 2381, 6416, 6437, 8776, 5221, 17936, 7241, 21352, 16441, 23440, 12721, 44936, 19657, 38096, 30281, 54536, 25261, 102992, 30785, 70216, 60065, 83536, 62957, 152456, 52061, 115856, 97621, 172072, 70645, 263056, 81401, 199240, 175997, 203536, 106081, 359816, 137601, 314512

Offset: 1

Laszlo Toth, Dec 28 2016

Inverse Moebius transform of A160891. - Seiichi Manyama, May 12 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013662, A060648, A064969, A160891, A280162, A344219, A344302, A344303.

with(numtheory):
# define Jordan totient function J(r,n)
J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
seq(add(J(4,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := With[{dd = Divisors[n]}, Sum[Times @@ EulerPhi @ {x, y, z, t} / EulerPhi[LCM[x, y, z, t]], {x, dd}, {y, dd}, {z, dd}, {t, dd}]];
Array[a, 50] (* Jean-François Alcover, Sep 28 2018 *)
f[p_, e_] := 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, x, sumdiv(n, y, sumdiv(n, z, sumdiv(n, t, eulerphi(x)*eulerphi(y)*eulerphi(z)*eulerphi(t)/eulerphi(lcm([x, y, z, t])))))); \\ Michel Marcus, Feb 26 2018

a160891(n) = sumdiv(n, d, moebius(n/d)*d^4)/eulerphi(n);
a(n) = sumdiv(n, d, a160891(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{a|n, b|n, c|n, d|n} phi(a)*phi(b)*phi(c)*phi(d)/phi(lcm(a, b, c, d)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.5010902655... . (End)
a(n) = Sum_{d divides n} J_4(d)/phi(d) = Sum_{1 <= i, j, k, l <= n} 1/phi(n/gcd(i,j,k,l,n)), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024

A293904 Decimal expansion of zeta(21).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 4, 7, 6, 9, 3, 2, 9, 8, 6, 7, 8, 7, 8, 0, 6, 4, 6, 3, 1, 1, 6, 7, 1, 9, 6, 0, 4, 3, 7, 3, 0, 4, 5, 9, 6, 6, 4, 4, 6, 6, 9, 4, 7, 8, 4, 9, 3, 7, 6, 0, 0, 2, 0, 7, 4, 8, 7, 3, 7, 6, 5, 9, 6, 8, 3, 9, 0, 8, 7, 8, 9, 8, 1, 5, 9, 8, 3, 3, 8, 7, 6, 6

Offset: 1

Frank Ellermann, Oct 19 2017

Web searches find 1.0000004769329867878 in Python tools. Simon Plouffe published 1000 digits for zeta(9) up to zeta(2051) many years ago.

			1.000000476932986787806...

Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each.
Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each. [Cached copy, with permission]
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each. (gzipped file)
Simon Plouffe, Identities inspired from Ramanujan Notebooks II, July 21, 1998.

Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678.

RealDigits[Zeta[21], 10, 100][[1]] (* Amiram Eldar, May 31 2021 *)

A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

Original entry on oeis.org

1, 29, 235, 925, 3101, 6815, 16759, 29597, 57097, 89929, 160931, 217375, 371125, 486011, 728735, 947101, 1419569, 1655813, 2475739, 2868425, 3938365, 4666999, 6435815, 6955295, 9690601, 10762625, 13874563, 15502075, 20510309, 21133315, 28628191, 30307229, 37818785

Offset: 1

Seiichi Manyama, May 18 2024

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

Cf. A068963, A084218, A372963.
Cf. A001160, A008683.
Cf. A013662, A013664, A088246.
Cf. A350156, A372964.
Cf. A371492, A372952.

f[p_, e_] := (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 5));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(4) = 2*Pi^2/21 = 0.939962323... (1/A088246). (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^4 * sigma_4(d^2)/sigma_2(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 2024

A367417 The squarefree kernels of the exponentially odd numbers: a(n) = A007947(A268335(n)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 2, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 6, 26, 3, 29, 30, 31, 2, 33, 34, 35, 37, 38, 39, 10, 41, 42, 43, 46, 47, 51, 53, 6, 55, 14, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 22, 89, 91, 93, 94, 95

Offset: 1

Amiram Eldar, Nov 17 2023

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

Cf. A007947, A268335, A367406.

Cf. A013662, A065463.

s[n_] := Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1], 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", "))); }

a(n) = A367406(n)/A268335(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 1.25661974314007532208..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

A369939 Numbers whose maximal exponent in their prime factorization is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A115063 at n = 2448. a(2448) = 2592 = 2^5 * 3^4 is not a term of A115063.
First differs from A209061 at n = 62.
Numbers k such that A051903(k) is a Fibonacci number.
The asymptotic density of this sequence is 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... .

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

Cf. A000045, A013662, A051903, A209061.
Subsequences: A005117, A062503, A062838, A113850, A115063.
Similar sequences: A368714, A369937, A369938.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}];
Select[Range[100], fibQ[Max[FactorInteger[#][[;; , 2]]]] &]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
is(n) = n == 1 || isfib(vecmax(factor(n)[, 2]));

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A351266 Sum of the cubes of the squarefree divisors of n.

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A351569 Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.

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A351600 a(n) = n^2 * Sum_{d^2|n} 1 / d^2.

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A352031 Sum of the cubes of the odd proper divisors of n.

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A116963 Inverse Moebius transform of the shifted tetrahedral numbers.

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A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

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A293904 Decimal expansion of zeta(21).

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A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.