A098198 -id:A098198 - cp's OEIS frontend

A371412 Euler totient function applied to the cubefull numbers (A036966).

1, 4, 8, 18, 16, 32, 54, 100, 64, 72, 162, 128, 294, 144, 256, 500, 216, 486, 288, 400, 512, 432, 1210, 576, 648, 800, 1024, 1458, 2028, 2058, 864, 1176, 2500, 1800, 1152, 1296, 1600, 2048, 4624, 2000, 1728, 2352, 1944, 4374, 6498, 2304, 2592, 3200, 4096, 5292, 4000

Offset: 1

Amiram Eldar, Mar 22 2024

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

Cf. A000010, A013661, A036966, A098198.

Similar sequences: A323333, A358039, A371413, A371414.

Join[{1}, EulerPhi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]
(* or *)
f[n_] := Module[{f = FactorInteger[n], p, e}, If[n == 1, 1, p = f[[;;, 1]]; e = f[[;;, 2]]; If[Min[e] > 2, Times @@ ((p-1) * p^(e-1)), Nothing]]]; Array[f, 20000]

lista(max) = {my(f); print1(1, ", "); for(k = 2, max, f = factor(k); if(vecmin(f[, 2]) > 2, print1(eulerphi(f), ", ")));}

a(n) = A000010(A036966(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p-1)^2*p)) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 3/p^4 + 1/p^5) = 1.65532418864085918623... .

A371414 Euler phi function applied to the cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 4, 18, 16, 100, 64, 72, 162, 294, 256, 288, 400, 1210, 648, 1024, 1458, 2028, 1176, 2500, 1800, 1152, 1600, 4624, 6498, 2592, 4096, 5292, 4840, 4704, 11638, 4608, 6400, 14406, 5832, 8112, 13122, 23548, 10000, 7200, 28830, 16200, 10368, 16384, 21780, 18496, 19360

Offset: 1

Amiram Eldar, Mar 22 2024

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

Cf. A000010, A098198, A335988.

Similar sequences: A323333, A371414, A371415.

Join[{1}, EulerPhi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

lista(max) = {my(f, ans); print1(1, ", "); for(k = 2, max, f = factor(k); ans = 1; for (i = 1, #f~, if (f[i, 2] == 1 || !(f[i, 2] % 2), ans = 0; break)); if(ans, print1(eulerphi(f), ", ")));}

a(n) = A000010(A335988(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p-1)^2*(p+1))) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 2/p^4) = 1.43921640806700099050... .

A070732 Size of largest conjugacy class in the group GL(2,Z_n).

Original entry on oeis.org

1, 3, 12, 12, 30, 36, 56, 48, 108, 90, 132, 144, 182, 168, 360, 192, 306, 324, 380, 360, 672, 396, 552, 576, 750, 546, 972, 672, 870, 1080, 992, 768, 1584, 918, 1680, 1296, 1406, 1140, 2184, 1440, 1722, 2016, 1892, 1584, 3240, 1656, 2256, 2304, 2744, 2250

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

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

Cf. A062354.
Cf. A000010, A000056.
Cf. A000082, A099892, A098198, A330523.

f[n_] := Block[{a = 1, b = FactorInteger[n]}, While[ Length[b] > 0, a = a*(b[[1, 1]] + 1)*b[[1, 1]]^(2b[[1, 2]] - If[ OddQ[ b[[1, 1]]], 1, 2]); b = Drop[b, 1]]; a]; Table[ f[n], {n, 1, 55}]
Table[n*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]/EulerPhi[2*n], {n, 1, 100}] (* Vaclav Kotesovec, Feb 01 2019 *)
f[p_, e_] := (p + 1)*p^(2*e - If[p == 2, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]+1)*f[i,1]^(2*f[i,2] - if(f[i,1]==2,2,1)));} \\ Amiram Eldar, Nov 05 2022

Multiplicative with a(p^e) = (p+1)*p^(2e - k), k = 1 if p is odd, k = 2 if p is 2.
a(n) = A000056(n)/A000010(2*n). - Vladeta Jovovic, Dec 22 2003
From R. J. Mathar, Apr 14 2011: (Start)
Dirichlet g.f.: (2^s-1)*zeta(s-1)*zeta(s-2)/((2^s+2)*zeta(2s-2)).
Dirichlet convolution of A000082 with a signed variant of A099892. (End)
Sum_{k=1..n} a(k) ~ 7*n^3 / (2*Pi^2). - Vaclav Kotesovec, Feb 01 2019
Sum_{n>=1} 1/a(n) = (13/11) * zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = (13/11) * A098198 * A330523 = 1.7136743536... . - Amiram Eldar, Nov 05 2022

Edited by Robert G. Wilson v, May 20 2002

A183031 Decimal expansion of Sum_{j>=1} tau(j)/j^4 = Pi^8/8100.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Dec 18 2010

This is the zeta-function Sum_{j>=1} A000005(j)/j^s evaluated at s=4. At s=2, we find A098198; at s=3, A183030.

Since tau(n)/n^4 is a multiplicative function, one finds an Euler product for the sum, which is expanded with an Euler transformation to a product of Riemann zeta functions as in A175639 for numerical evaluation.

			1.1714235822309350626084... = 1 + 2/2^4 + 2/3^4 + 3/4^4 + 2/5^4 + 4/6^4 + 2/7^4 + ...

Index entries for transcendental numbers

Cf. A000005, A098198, A183030, A175639.

Maple
```
evalf(Pi^8/8100) ;
```

RealDigits[Zeta[4]^2, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

zeta(4)^2 \\ Charles R Greathouse IV, Mar 04 2015

Equals the Euler product Product_{p prime} (1 + (2*p^s - 1)/(p^s - 1)^2) at s=4, which is the square of A013662.

A227929 Decimal expansion of 36/Pi^4.

Original entry on oeis.org

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

Offset: 0

Arkadiusz Wesolowski, Oct 09 2013

Ernesto Cesaro asserted that lim n -> infinity A002321(n)/n = 36/Pi^4 using a fallacious argument. In fact this limit equals zero.

			36/Pi^4 = 0.369575361168636066809500197....

Ernesto Cesaro, Sur diverses questions d'arithmetique. Mem. Soc. Roy. Sci. Liege 10 (1883), 1-350. Reprinted in Opere Scelte I, Vol. 1, pp. 10-362.
Wladyslaw Narkiewicz, The development of prime number theory: from Euclid to Hardy and Littlewood, Springer-Verlag, New York, 2000, p. 31.

Index entries for transcendental numbers

Cf. A000796, A002321.

pi:=Pi(RealField(107)); Reverse(Intseq(Floor(10^105*36/pi^4)));

Mathematica
```
RealDigits[N[36/Pi^4, 105]][[1]]
```

default(realprecision, 105); x=360/Pi^4; for(n=1, 105, d=floor(x); x=(x-d)*10; print1(d, ", "));

Equals Product_{primes p} (1 - 2/p^2 + 1/p^4). - Vaclav Kotesovec, Jun 20 2020

Equals 1/A098198. - R. J. Mathar, Jul 21 2025

A386403 Decimal expansion of zeta(3)/3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 20 2025

			0.4006856343865314284665793871704833...

Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 424.

Cf. A002117, A072691 (zeta(2)/2), A098198 (zeta(4)/4), A386404 (zeta(5)/5), A259928 (zeta(6)/6).

Maple
```
Digits := 100 ; Zeta(3.0)/3. ;
```

RealDigits[Zeta[3]/3, 10 , 120][[1]] (* Amiram Eldar, Jul 21 2025 *)

PARI
```
zeta(3)/3 \\ Amiram Eldar, Jul 21 2025
```

Equals A002117/3.

Equals Sum_{k>=1} cos(k*Pi/3)/k^3 (Shamos, 2011). - Amiram Eldar, Jul 21 2025

A098199 Continued fraction expansion for Pi^4/36, the limiting value of A024916(n)/A002088(n).

Original entry on oeis.org

2, 1, 2, 2, 1, 1, 46, 459, 2, 3, 1, 2, 2, 1, 1, 8, 18, 1, 1, 1, 1, 6, 5, 6, 14, 1, 2, 1, 1, 140, 1, 2, 1, 1, 2, 1, 9, 16, 1, 2, 1, 1, 1, 15, 1, 3, 55, 1, 1, 12, 1, 1, 5, 4, 6, 13, 2, 2, 7, 2, 32, 1, 1, 6, 1, 1, 54, 1, 1, 1, 21, 1, 2, 1, 3, 4, 5, 15, 1, 6, 1, 2, 5, 1, 1, 7, 1, 834, 2, 1, 4, 8, 3, 2, 3, 1, 5

Offset: 0

Labos Elemer, Sep 21 2004

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A024916, A002088, A098198 (decimal expansion).

Mathematica
```
ContinuedFraction[(Pi^4)/36, 256]
```

contfrac(Pi^4/36) \\ Amiram Eldar, Mar 08 2025

Offset changed by Andrew Howroyd, Aug 04 2024

A338106 Decimal expansion of Sum_{m>1, n>1} 1/(m^2*n^2-1).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Oct 10 2020

For p>1, q>1 in R, Sum_{m >1, n>1} 1/(m^p*n^q-1) = Sum_{k>0} (zeta(k*p) - 1) * (zeta(k*q) - 1) [Proof in References]. This sequence corresponds to p = q = 2.

Double inequality: Sum_{m>1, n>1} 1/(m^2*n^2+1) = A338107 = 0.409... < Sum_{m>1, n>1} 1/(m^2*n^2) = (zeta(2)-1)^2 = 0.415... < Sum_{m>1, n>1} 1/(m^2*n^2-1) = this constant = 0.423...

			0.4230355257613131597420971016391038628995464... (with help of _Amiram Eldar_).

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.25, p. 277.

Cf. A098198, A333972, A338107.

RealDigits[Sum[(Zeta[2*k] - 1)^2, {k, 1, 100}], 10, 90][[1]] (* Amiram Eldar, Oct 10 2020 *)

sumpos(k=1, (zeta(2*k) - 1)^2) \\ Michel Marcus, Oct 10 2020

Equals Sum_{k>0} (zeta(2*k) - 1)^2.

Equals -3/4 + Sum_{k>=2} (1/2 - Pi*cot(Pi/k)/(2*k)). - Vaclav Kotesovec, Oct 14 2020

A338107 Decimal expansion of Sum_{m>1, n>1} 1/(m^2*n^2+1).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Oct 10 2020

Double inequality: Sum_{m>1, n>1} 1/(m^2*n^2+1) = this constant = 0.409... < Sum_{m>1, n>1} 1/(m^2*n^2) = (zeta(2)-1)^2 = 0.415... < Sum_{m>1, n>1} 1/(m^2*n^2-1) = A338106 = 0.423...

			0.40944792489076040575341901269025385039506836638... (with help of _Amiram Eldar_).

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.25, p. 277.

Cf. A098198, A333972, A338106.

RealDigits[Sum[(-1)^(k - 1)*(Zeta[2*k] - 1)^2, {k, 1, 100}], 10, 90][[1]] (* Amiram Eldar, Oct 10 2020 *)

sumalt(k=1, (-1)^(k-1) * (zeta(2*k) - 1)^2) \\ Michel Marcus, Oct 10 2020

Equals Sum_{k>0} (-1)^(k-1) * (zeta(2*k) - 1)^2.

Equals 3/2 - Pi*coth(Pi) + Sum_{k>=1} (Pi*coth(Pi/k)/(2*k) - 1/2). - Vaclav Kotesovec, Oct 14 2020

A364488 Decimal expansion of zeta(2) * primezeta(2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jul 26 2023

			0.743917187869767974935961806463534512710431875022875115314346546...

Eric Weisstein's World of Mathematics, Prime Zeta Function.
Eric Weisstein's World of Mathematics, Riemann Zeta Function zeta(2).

Cf. A001221, A013661, A085548, A098198, A364490.

RealDigits[Zeta[2] PrimeZetaP[2], 10, 105][[1]]

zeta(2) * sumeulerrat(1/p, 2) \\ Amiram Eldar, Jul 28 2023

Equals Sum_{k>=1} omega(k) / k^2, where omega(k) is the number of distinct primes dividing k (A001221).

cp's OEIS Frontend

A371412 Euler totient function applied to the cubefull numbers (A036966).

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A371414 Euler phi function applied to the cubefull exponentially odd numbers (A335988).

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A070732 Size of largest conjugacy class in the group GL(2,Z_n).

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A183031 Decimal expansion of Sum_{j>=1} tau(j)/j^4 = Pi^8/8100.

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A227929 Decimal expansion of 36/Pi^4.

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A386403 Decimal expansion of zeta(3)/3.

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A098199 Continued fraction expansion for Pi^4/36, the limiting value of A024916(n)/A002088(n).

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