A330523 -id:A330523 - cp's OEIS frontend

A361264 Multiplicative with a(p^e) = p^(e + 2), e > 0.

1, 8, 27, 16, 125, 216, 343, 32, 81, 1000, 1331, 432, 2197, 2744, 3375, 64, 4913, 648, 6859, 2000, 9261, 10648, 12167, 864, 625, 17576, 243, 5488, 24389, 27000, 29791, 128, 35937, 39304, 42875, 1296, 50653, 54872, 59319, 4000, 68921, 74088, 79507, 21296, 10125

Offset: 1

Vaclav Kotesovec, Mar 06 2023

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

Cf. A000005, A003557, A064549, A065483, A007947, A330523, A360997, A361266.

g[p_, e_] := p^(e+2); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, 1 + p^3*X/(1 - p*X))[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + p^3/(p^s - p)).
Dirichlet g.f.: zeta(s-3) * zeta(s-1) * Product_{primes p} (1 + p^(4-2*s) - p^(6-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * zeta(3) * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.53589615382833799980850263131854595064822237...
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = n * A007947(n)^2 = A064549(n) * A007947(n) = A064549(A064549(n)).
A000005(a(n)) = A360997(n).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) = A065483. (End)

A114810 Number of complex, weakly primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 2, 4, 4, 10, 2, 12, 6, 8, 4, 16, 4, 18, 4, 12, 10, 22, 4, 16, 12, 12, 6, 28, 8, 30, 8, 20, 16, 24, 4, 36, 18, 24, 8, 40, 12, 42, 10, 16, 22, 46, 8, 36, 16, 32, 12, 52, 12, 40, 12, 36, 28, 58, 8, 60, 30, 24, 16, 48, 20, 66, 16, 44, 24, 70, 8, 72, 36, 32, 18, 60, 24, 78

Offset: 1

Steven Finch, Feb 19 2006

Any primitive Dirichlet character is weakly primitive (not conversely). Jager uses the phrase "proper character", but this conflicts with other authors (e.g., W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, p. 224) who use the word "proper" to mean the same as "primitive".

Equals Mobius transform of A055653. - Gary W. Adamson, Feb 28 2009

			The function chi defined on the integers by chi(1)=1, chi(5)=-1 and chi(2)=chi(3)=chi(4)=chi(6)=0 [and extended periodically] is a weakly primitive character mod 6, but not mod 12 or mod 18. In this sense, we eliminate the "overcounting" of complex Dirichlet characters in A000010.

Amiram Eldar, Table of n, a(n) for n = 1..10000
H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973) 452-455.

Cf. A000010, A007431, A055231, A330523.

Cf. A055653. [Gary W. Adamson, Feb 28 2009]

b[n_] := Sum[EulerPhi[d]*MoebiusMu[n/d], {d, Divisors[n]}]; squareFreeKernel[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := Sum[b[n/d], {d, Divisors[Denominator[n/squareFreeKernel[n]^2]]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Sep 07 2015 *)
f[p_, e_] := If[e == 1, p - 1, (p - 1)^2*p^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2022 *)

a(n) is multiplicative with a(p) = phi(p), a(p^k) = phi(p^k)-phi(p^(k-1)) and phi(n) = A000010(n).
a(n) = Sum_{d} A007431(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 / 2 = 0.2679480769... . - Amiram Eldar, Nov 04 2022

A175639 Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

Equals (49/64)*(668/729)*(15304/15625)*(116724/117649)*... inserting p= A000040 = 2, 3, 5, 7.. into the factor. Slightly larger than Product_{p=primes} (1-3/p^3) = 0.534566872085103888416775...

			0.678234491917391978035...

Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 85.
Takashi Taniguchi, A mean value theorem for the square of class number times regulator of quadratic extensions, Annales de l'institut Fourier. Vol. 58, No. 2 (2008), pp. 625-670; arXiv preprint, arXiv:math/0410531 [math.NT], 2004-2006.
Eric Weisstein's World of Mathematics, Prime Products.
Eric Weisstein's World of Mathematics, Taniguchi's Constant.
Wikipedia, Euler Product.

Cf. A256392, A330523.

read("transforms") : efact := 1-3/p^3+2/p^4+1/p^5-1/p^6 ; Digits := 130 : tm := 310 : subs (p=1/x,1/efact) ; taylor(%,x=0,tm) : L := [seq(coeftayl(%,x=0,i),i=1..tm-1)] : Le := EULERi(L) : x := 1.0 :
for i from 2 to nops(Le) do x := x/evalf(Zeta(i))^op(i,Le) ; x := evalf(x) ; print(x) ; end do:

digits = 105; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s];
LR = LinearRecurrence[{0, 0, 3, -2, -1, 1}, {0, 0, -9, 8, 5, -33}, 2 m0];
r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m-dm], 10, digits][[1]], Print[m]; m = m+dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-3/p^3+2/p^4+1/p^5-1/p^6) \\ Amiram Eldar, Mar 04 2021

More digits from Jean-François Alcover, Apr 15 2016

A318661 Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 3, 19, 5, 11, 3, 13, 7, 15, 5, 17, 19, 19, 5, 21, 11, 23, 9, 59, 13, 95, 7, 29, 15, 31, 9, 33, 17, 35, 19, 37, 19, 39, 15, 41, 21, 43, 11, 95, 23, 47, 15, 123, 59, 51, 13, 53, 95, 55, 21, 57, 29, 59, 15, 61, 31, 133, 67, 65, 33, 67, 17, 69, 35, 71, 57, 73, 37, 177, 19, 77, 39, 79, 25, 2019, 41, 83, 21, 85, 43, 87, 33, 89

Offset: 1

Antti Karttunen, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A055653, A318662 (denominators).

up_to = 1+(2^16);
A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA055653(n)));
A318661(n) = numerator(v318661_62[n]);
A318662(n) = denominator(v318661_62[n]);
A318663(n) = valuation(A318662(n),2);

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1 + X^2 - p*X^2 - X)/((1-X)*(1-p*X)))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 10 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A055653(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025: (Start)
Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 1/p^(2*s-1) - 1/p^s).
Sum_{k=1..n} A318661(k) / A318662(k) ~ n^2 * sqrt(Pi*f(2)/(24*log(n))) * (1 - ((gamma - 1)/2 + f'[2]/(2*f(2)) + 3*zeta'(2)/Pi^2) / (2*log(n))), where
f(2) = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.5358961538283379998085026313185459506482223745141452711510108346133288119...
f'(2)/f(2) = Sum_{primes p} (p^2 + 2*p - 2) * log(p) / (p^4 - p^2 - p + 1) = 0.8249574883141571786856463180997569604486048593127391054584235479395133668...
and gamma is the Euler-Mascheroni constant A001620. (End)

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

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/(zeta(2*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

			1.8107814761215629522431259...

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Mathematica
```
RealDigits[N[5005/2764,105]][[1]]
```

default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.25(3617945007235890014471780028943560057887120115774240231548480463096960
9261939218523878437047756874095513748191027496382054992764109985528219
9710564399421128798842257597684515195369030390738060781476121562952243
12590448625180897250).

			1.25361794500723589001447178...

Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340065.

Mathematica
```
RealDigits[N[3465/2764, 105]][[1]]
```

default(realprecision, 105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

Equals 3465/2764 = 3^2*5*7*11/(2^2*691).
Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.
Equals (9/13)*A340065.

A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

Original entry on oeis.org

1, 6, 12, 30, 30, 72, 56, 138, 123, 180, 132, 360, 182, 336, 360, 602, 306, 738, 380, 900, 672, 792, 552, 1656, 795, 1092, 1176, 1680, 870, 2160, 992, 2538, 1584, 1836, 1680, 3690, 1406, 2280, 2184, 4140, 1722, 4032, 1892, 3960, 3690, 3312, 2256, 7224, 2835, 4770, 3672, 5460

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A000385, A034761, A038040, A062354, A062952, A183699, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 12 2022 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/3) * A183699 * A330523 = 0.581007... . (End)

A341638 a(n) = Sum_{d|n} phi(d) * sigma(d) * tau(n/d).

Original entry on oeis.org

1, 5, 10, 23, 26, 50, 50, 101, 97, 130, 122, 230, 170, 250, 260, 427, 290, 485, 362, 598, 500, 610, 530, 1010, 671, 850, 904, 1150, 842, 1300, 962, 1761, 1220, 1450, 1300, 2231, 1370, 1810, 1700, 2626, 1682, 2500, 1850, 2806, 2522, 2650, 2210, 4270, 2493, 3355, 2900, 3910, 2810, 4520

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Inverse Moebius transform of A062952.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A007430, A062354, A062952, A191831.

Cf. A002117, A013661, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 54}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 54}]
f[p_, e_] := (p^(2*e + 4) - p^(e + 3) - 2*p^(e + 2) - p^(e + 1) + (e + 1)*p^3 - (e - 1)*p + 1)/(p^2 - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 26 2023 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * sigma(n/gcd(n,k)).
a(n) = Sum_{d|n} A062952(d).
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*sigma(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021
From Amiram Eldar, Jan 26 2023: (Start)
Multiplicative with a(p^e) = (p^(2*e+4) - p^(e+3) - 2*p^(e+2) - p^(e+1) + (e+1)*p^3 - (e-1)*p + 1)/(p^2-1)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)^2/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A013661 * A002117^2 * A330523 / 3 = 0.424578... . (End)

A386624 a(n) = Sum_{d|n} sigma(d) * phi(d) * mu(n/d).

Original entry on oeis.org

1, 2, 7, 11, 23, 14, 47, 46, 70, 46, 119, 77, 167, 94, 161, 188, 287, 140, 359, 253, 329, 238, 527, 322, 596, 334, 642, 517, 839, 322, 959, 760, 833, 574, 1081, 770, 1367, 718, 1169, 1058, 1679, 658, 1847, 1309, 1610, 1054, 2207, 1316, 2346, 1192, 2009, 1837, 2807, 1284, 2737, 2162, 2513, 1678, 3479, 1771, 3719, 1918, 3290, 3056, 3841, 1666, 4487, 3157, 3689

Offset: 1

Wesley Ivan Hurt, Jul 27 2025

Möbius transform of sigma(n) * phi(n) = A062354(n).

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

Cf. A000010 (phi), A000203 (sigma), A062354, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 100}]
f[p_, e_] := p^(2*e) - p^(e-1) - If[e > 1, p^(2*e-2) - p^(e-2), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 27 2025 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i, 2];  p^(2*e) - p^(e - 1) - if(e > 1, p^(2*e - 2) - p^(e - 2), 1));} \\ Amiram Eldar, Jul 27 2025

a(n) = Sum_{d|n} A062354(d) * mu(n/d).
From Amiram Eldar, Jul 27 2025: (Start)
Multiplicative with a(p) = p^2 - 2, and a(p^e) = p^(2*e) - p^(2*e-2) - p^(e-1) + p^(e-2) for e >= 2.
Dirichlet g.f.: (zeta(s-2) * zeta(s-1) / zeta(s)) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s + 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/(3*zeta(3))) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523*zeta(2)/(3*zeta(3)) = 0.24444595409976589792... . (End)

cp's OEIS Frontend

A361264 Multiplicative with a(p^e) = p^(e + 2), e > 0.

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A114810 Number of complex, weakly primitive Dirichlet characters modulo n.

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A175639 Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).

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A318661 Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

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

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A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

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