A013662 -id:A013662 - cp's OEIS frontend

A231535 Decimal expansion of Pi^4/15.

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

Offset: 1

Stanislav Sykora, Nov 12 2013

Under proper scaling, the radiation distribution density function in terms of frequency is given by prl(x) = x^3/(exp(x)-1), the Planck's radiation law. This constant is the integral of prl(x) from 0 to infinity and leads to the total amount of electromagnetic radiation emitted by a body.

Also, in an 8-dimensional unit-radius hypersphere, equals one-fifth of its surface (A164109), and twice the integral of r^2 over its volume.

			6.4939394022668291490960221792470074166485057115123614460978572926647...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Ilham A. Aliev, Ayhan Dil, Tornheim-like series, harmonic numbers and zeta values, arXiv:2008.02488 [math.NT], 2020, p. 4.
Stanislav Sykora, Surface Integrals over n-Dimensional Spheres
Stanislav Sykora, Volume Integrals over n-Dimensional Ellipsoids
Wikipedia, Planck's Law
Ke Xiao, Dimensionless Constants and Blackbody Radiation Laws, Electronic Journal of Theoretical Physics, 8(2011), 379-388, Eq.6.
Index entries for transcendental numbers

Cf. A000796, A013662, A081819, A164109.

RealDigits[Pi^4/15, 10, 105][[1]] (* Bruno Berselli, Nov 12 2013 *)

PARI
```
Pi^4/15 \\ Michel Marcus, Aug 07 2020
```

6*prodeulerrat(p^4/(p^4-1)) \\ Hugo Pfoertner, Aug 07 2020

Equals 6*zeta(4), see A013662. - Bruno Berselli, Nov 12 2013
Equals Gamma(4)*zeta(4) = 2*3*Product_{prime p} (p^4/(p^4-1)). - Stanislav Sykora, Oct 20 2014
Equals Sum_{n, m >= 1} H(n+m)/(n*m*(n+m)) where H(n) is the n-th harmonic number. See Aliev and Dil. - Michel Marcus, Aug 07 2020
From Amiram Eldar, Aug 14 2020: (Start)
Equals Integral_{x=0..oo} x^3/(exp(x)-1) dx.
Equals Integral_{x=0..1} log(x)^3/(x-1) dx. (End)
Equals psi'''(1), the third derivative of the digamma function at 1. - R. J. Mathar, Aug 29 2023

A240976 Decimal expansion of 3zeta(3)/(2Pi^2), a constant appearing in the asymptotic evaluation of the average LCM of two integers chosen independently from the uniform distribution [1..n].

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 07 2014

15*zeta(3)/Pi^2 = 10 * (this constant) equals the asymptotic mean of the abundancy index of the squares (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			0.18269074235035962468150918282692865988200290126984361751783...

Persi Diaconis and Paul Erdős, On the distribution of the greatest common divisor, Technical Report No. 12 (1977) U.S. Army Research Office.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 17.
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (4).
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv:1310.7053 [math.NT], 2013-2014, formula (47), p. 23.

Cf. A000188, A002117, A013662, A048250, A059956.

RealDigits[3*Zeta[3]/(2*Pi^2), 10, 102] // First

Equals zeta(3)/(4*zeta(2)) = 3*zeta(3)/(2*Pi^2).
From Amiram Eldar, Jan 25 2024: (Start)
Equals (1/10) * Sum_{k>=1} A000188(k)/k^2.
Equals (1/10) * Sum_{k>=1} A048250(k)/k^3. (End)

A248227 Least k such that zeta(4) - sum{1/h^4, h = 1..k} < 1/n^3.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 46, 47

Offset: 1

Clark Kimberling, Oct 05 2014

This sequence and A248230 provide insight into the manner of convergence of sum{1/h^4, h = 0..k}. Since a(n+1) - a(n) is in {0,1} for n >= 1, A248228 and A248229 are complementary.

			Let s(n) = sum{1/h^4, h = 1..n}.  Approximations are shown here:
n ... zeta(4) - s(n) ... 1/n^3
1 ... 0.0823232 .... 1
2 ... 0.0198232 .... 0.125
3 ... 0.0074775 .... 0.037
4 ... 0.0035713 .... 0.015
5 ... 0.0019713 .... 0.008
6 ... 0.0011997 .... 0.005
a(6) = 4 because zeta(4) - s(4) < 1/216 < zeta(4) - s(3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A248228, A248229, A248230, A013662.

$MaxExtraPrecision = Infinity; z = 400; p[k_] := p[k] = Sum[1/h^4, {h, 1, k}];
N[Table[Zeta[4] - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Zeta[4] - p[#] < 1/n^3 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]   (* A248227 *)
Flatten[Position[Differences[u], 0]]  (* A248228 *)
Flatten[Position[Differences[u], 1]]  (* A248229 *)
f = Table[Floor[1/(Zeta[4] - p[n])], {n, 1, z}]  (* A248230 *)

a(n) ~ 3^(-1/3) * n. - Vaclav Kotesovec, Oct 09 2014

A282213 Coefficients in q-expansion of (E_2^3E_4 - 3E_2^2E_6 + 3E_2E_4^2 - E_4E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 72, 756, 4672, 15750, 54432, 117992, 299520, 551853, 1134000, 1772892, 3532032, 4829006, 8495424, 11907000, 19173376, 24142482, 39733416, 47052740, 73584000, 89201952, 127648224, 148048056, 226437120, 246109375, 347688432, 402320520, 551258624, 594847710

Offset: 0

Seiichi Manyama, Feb 09 2017

Multiplicative because A001158 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^6*6^3 + 2^6*3^3 + 3^6*2^3 + 6^6*1^3 = 54432.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282211 (phi_{4, 3}), this sequence (phi_{6, 3}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282586 (E_2^3*E_4), A282595 (E_2^2*E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), this sequence (n^3*sigma_3(n))
Cf. A013662.

terms = 30;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E2[x]^3*E4[x] - 3 E2[x]^2*E6[x] + 3 E2[x] E4[x]^2 - E4[x] E6[x])/3456 + O[x]^terms // CoefficientList[#, x]&
(* or: *)
Table[n^3*DivisorSigma[3, n], {n, 0, terms-1}] (* Jean-François Alcover, Feb 27 2018 *)
nmax = 30; CoefficientList[Series[Sum[k^6*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if (n, n^3*sigma(n, 3), 0); \\ Michel Marcus, Feb 27 2018

G.f.: phi_{6, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282586(n) - 3*A282595(n) + 3*A282101(n) - A013974(n))/3456. - Seiichi Manyama, Feb 19 2017
a(n) = n^3*A001158(n) for n > 0. - Seiichi Manyama, Feb 19 2017
Sum_{k=1..n} a(k) ~ zeta(4) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(3*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-6). (End)
G.f.: Sum_{k>=1} k^6*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - Vaclav Kotesovec, Aug 02 2025

A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

Original entry on oeis.org

0, 1, 4, 11, 20, 40, 56, 95, 124, 186, 220, 336, 364, 512, 584, 775, 816, 1129, 1140, 1526, 1600, 1992, 2024, 2720, 2620, 3290, 3400, 4176, 4060, 5280, 4960, 6231, 6208, 7362, 7216, 9195, 8436, 10280, 10248, 12270, 11480, 14432, 13244, 16192, 15884, 18240

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A000292, A032741, A065608, A069153, A363605, A363606.
Cf. A059358, A363607, A363611.
Cf. A000203, A001158, A013662, A092348.

a[n_] := (DivisorSigma[3, n] - DivisorSigma[1, n])/6; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^4)))

a(n) = my(f = factor(n)); (sigma(f, 3) - sigma(f))/6; \\ Amiram Eldar, Dec 30 2024

a(n) = (sigma_3(n) - sigma(n))/6 = A092348(n)/6.
G.f.: Sum_{k>0} binomial(k+1,3) * x^k/(1 - x^k).
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) - zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).

Original entry on oeis.org

1, 15, 79, 239, 621, 1185, 2395, 3823, 6397, 9315, 14631, 18881, 28549, 35925, 49059, 61167, 83505, 95955, 130303, 148419, 189205, 219465, 279819, 302017, 388121, 428235, 518155, 572405, 707253, 735885, 923491, 978671, 1155849, 1252575, 1487295, 1528883

Offset: 1

Seiichi Manyama, May 18 2024

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

Column k=3 of A372968.
Cf. A001159, A008683.
Cf. A013662, A013663.
Cf. A371492, A372962.

f[p_, e_] := (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1); 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*sigma(d,4));

a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_4(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1).
Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(5)/zeta(4) = 0.958057374... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d) * 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, x_2, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 2024

A015634 Number of ordered quadruples of integers from [ 1..n ] with no global factor.

Original entry on oeis.org

1, 4, 13, 29, 63, 106, 189, 289, 444, 626, 911, 1203, 1657, 2130, 2766, 3462, 4430, 5359, 6688, 7992, 9670, 11405, 13704, 15840, 18730, 21548, 25037, 28521, 33015, 37067, 42522, 47690, 53940, 60108, 67760, 74748, 83886, 92433, 102629, 112469, 124809, 135763, 149952

Offset: 1

Olivier Gérard

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Column k=4 of A177976.
Partial sums of A117108.
Cf. A000332, A002088, A013662, A015631, A015650.

a[n_] := Sum[DivisorSum[k, MoebiusMu[k/#]*Binomial[# + 2, 3] &], {k, 1, n}]; Array[a, 45] (* Amiram Eldar, Jun 07 2025 *)

a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+2, 3))); \\ Seiichi Manyama, Jun 12 2021

a(n) = binomial(n+3, 4)-sum(k=2, n, a(n\k)); \\ Seiichi Manyama, Jun 12 2021

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^4)/(1-x)) \\ Seiichi Manyama, Jun 12 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A015634(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A015634(k1)
        j, k1 = j2, n//j2
    return n*(n+1)*(n+2)*(n+3)//24-c+j-n # Chai Wah Wu, Apr 18 2021

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^4. - Ilya Gutkovskiy, Feb 14 2020
a(n) = n*(n+1)*(n+2)*(n+3)/24 - Sum_{j=2..n} a(floor(n/j)) = A000332(n+3) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Apr 18 2021
a(n) ~ n^4 / (24*zeta(4)). - Amiram Eldar, Jun 08 2025

A030140 The nonsquares squared.

Original entry on oeis.org

4, 9, 25, 36, 49, 64, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025

Offset: 1

N. J. A. Sloane

The complement of the fourth powers A000583 within the squares A000290. - Peter Munn, Aug 20 2019

			a(1)=2^2, a(2)=3^2, a(3)=5^2, a(4)=6^2, a(5)=7^2, ..., a(n)=(integer which is not a perfect square)^2.

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

Cf. A000037, A000290, A000583, A013661, A013662, A062503.
Positions of 2's in A352080.
Related to A016945 via A225546.

[(n + Floor(1/2 + Sqrt(n)))^2: n in [1..60]]; // Vincenzo Librandi, Apr 06 2020

a:=proc(n) if type(sqrt(n),integer)=false then n^2 else fi end: seq(a(n),n=1..70); # Emeric Deutsch, Apr 11 2007

a[n_] := (n + Floor[1/2 + Sqrt[n]])^2;
Array[a, 50] (* Jean-François Alcover, Apr 05 2020 *)

from math import isqrt
def A030140(n): return (n+(k:=isqrt(n))+int(n>=k*(k+1)+1))**2 # Chai Wah Wu, Jun 17 2024

a(n) = A000037(n)^2.
Sum_{n>=1} 1/a(n) = zeta(2) - zeta(4) = A013661 - A013662 = 0.5626108331... - Amiram Eldar, Nov 14 2020
{a(n) : n >= 1} = {A225546(6m+3) : m >= 0}. - Peter Munn, Nov 17 2022

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A052277 a(n) = (4n+2)!/2^(2n+1).

Original entry on oeis.org

1, 90, 113400, 681080400, 12504636144000, 548828480360160000, 49229914688306352000000, 8094874872198213459360000000, 2252447502438386084347676160000000, 997586474354936812896742294502400000000, 669959124447288464805194190141921792000000000

Offset: 0

N. J. A. Sloane, Feb 05 2000

J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, arXiv:hep-th/9611004, 1996.
Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.

Cf. A002432 (denominators of zeta(2*n)/Pi^(2*n)).

Cf. A068447, A067912, A013662 (zeta(4)).

Table[(4n+2)!/2^(2n+1), {n, 0, 10}] (* Amiram Eldar, Feb 25 2022 *)

a(n) = (4*n+2)!/2^(2*n+1); \\ Michel Marcus, Feb 20 2022

sin(x)*sinh(x) = Sum_{n>=0} (-1)^n*x^(4n+2)/a(n). - Benoit Cloitre, Feb 02 2002
a(n) = Pi^(4n)/Zeta({4}n) where ({4}_n) is the standard multiple zeta values notation for (4, ..., 4) where the multiplicity of 4 is n. - _Roudy El Haddad, Feb 19 2022
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=0} 1/a(n) = (cosh(sqrt(2)) - cos(sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = sin(1)*sinh(1). (End)

A183700 Decimal expansion of zeta(3)*zeta(4), the product of two Riemann zeta values.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 06 2011

Equals the Dirichlet zeta-function sum_{n>=1} A000203(n)/n^s at s=4.

			Equals 1.301014114532488574154...

Cf. A183699 (s=3).

Maple
```
evalf(Zeta(3)*Zeta(4)) ;
```

RealDigits[N[Zeta[3] * Zeta[4], 75]][[1]] (* Alonso del Arte, Jan 06 2011 *)

Equals A002117 * A013662.

cp's OEIS Frontend

A231535 Decimal expansion of Pi^4/15.

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A240976 Decimal expansion of 3*zeta(3)/(2*Pi^2), a constant appearing in the asymptotic evaluation of the average LCM of two integers chosen independently from the uniform distribution [1..n].

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A248227 Least k such that zeta(4) - sum{1/h^4, h = 1..k} < 1/n^3.

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A282213 Coefficients in q-expansion of (E_2^3*E_4 - 3*E_2^2*E_6 + 3*E_2*E_4^2 - E_4*E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

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A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).

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A015634 Number of ordered quadruples of integers from [ 1..n ] with no global factor.

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A240976 Decimal expansion of 3zeta(3)/(2Pi^2), a constant appearing in the asymptotic evaluation of the average LCM of two integers chosen independently from the uniform distribution [1..n].

A282213 Coefficients in q-expansion of (E_2^3E_4 - 3E_2^2E_6 + 3E_2E_4^2 - E_4E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.