A127473 -id:A127473 - cp's OEIS frontend

A029939 a(n) = Sum_{d|n} phi(d)^2.

1, 2, 5, 6, 17, 10, 37, 22, 41, 34, 101, 30, 145, 74, 85, 86, 257, 82, 325, 102, 185, 202, 485, 110, 417, 290, 365, 222, 785, 170, 901, 342, 505, 514, 629, 246, 1297, 650, 725, 374, 1601, 370, 1765, 606, 697, 970, 2117, 430, 1801, 834, 1285, 870, 2705, 730, 1717, 814, 1625

Offset: 1

N. J. A. Sloane

Equals the inverse Mobius transform (A051731) of A127473. - Gary W. Adamson, Aug 20 2008

Number of (i,j) in {1,2,...,n}^2 such that gcd(n,i) = gcd(n,j). - Benoit Cloitre, Dec 31 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051731, A062367, A065463, A127473, A253905.

with(numtheory): A029939 := proc(n) local i,j; j := 0; for i in divisors(n) do j := j+phi(i)^2; od; j; end;
# alternative
N:= 1000: # to get a(1)..a(N)
A:= Vector(N,1):
for d from 2 to N do
  pd:= numtheory:-phi(d)^2;
  md:= [seq(i,i=d..N,d)];
  A[md]:= map(`+`,A[md],pd);
od:
seq(A[i],i=1..N); # Robert Israel, May 30 2016

Table[Total[EulerPhi[Divisors[n]]^2],{n,60}] (* Harvey P. Dale, Feb 04 2017 *)
f[p_, e_] := (p^(2*e)*(p-1)+2)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = sumdiv(n, d, eulerphi(d)^2); \\ Michel Marcus, Jan 17 2017

Multiplicative with a(p^e) = (p^(2*e)*(p-1)+2)/(p+1). - Vladeta Jovovic, Nov 19 2001
G.f.: Sum_{k>=1} phi(k)^2*x^k/(1 - x^k), where phi(k) is the Euler totient function (A000010). - Ilya Gutkovskiy, Jan 16 2017
a(n) = Sum_{k=1..n} phi(n/gcd(n, k)). - Ridouane Oudra, Nov 28 2019
Sum_{k>=1} 1/a(k) = 2.3943802654751092440350752246012273573942903149891228695146514601814537713... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/(3*zeta(2))) * Product_{p prime} (1 - 1/(p*(p+1))) = A253905 * A065463 / 3 = 0.171593... . - Amiram Eldar, Oct 25 2022

A057434 a(n) = Sum_{k=1..n} phi(k)^2.

Original entry on oeis.org

1, 2, 6, 10, 26, 30, 66, 82, 118, 134, 234, 250, 394, 430, 494, 558, 814, 850, 1174, 1238, 1382, 1482, 1966, 2030, 2430, 2574, 2898, 3042, 3826, 3890, 4790, 5046, 5446, 5702, 6278, 6422, 7718, 8042, 8618, 8874, 10474, 10618, 12382, 12782

Offset: 1

N. J. A. Sloane, Sep 08 2000

nonn

Partial sums of A127473. - R. J. Mathar, Sep 29 2008

Ivan Neretin, Table of n, a(n) for n = 1..10000
U. Balakrishnan & Y.-F. S. Pétermann, The Dirichlet series of zeta(s)*zeta(s+1)^alpha*f(s+1): On an error term associated with its coefficients, Acta Arith. 75 (1996), 39--69.

Cf. A000010, A002088, A061502, A072379, A074789.

FoldList[Plus, 1, EulerPhi[Range[2, 50]]^2] (* Ivan Neretin, May 30 2015 *)

a(n) = sum(k=1, n, eulerphi(k)^2); \\ Michel Marcus, Dec 20 2015

We can derive an asymptotic formula from a general formula given in the reference, namely: a(n) = C*n^3 + O(log(x)^(4/3)log(log(x))^(8/3)) where C = (1/3)/zeta(2)^2*Product_{p prime}(1+1/(p-1)/(p+1)^2) = 0.142749835225698(...). - Benoit Cloitre, Dec 22 2015

a(n) ~ c * n^3 / 3, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319... - Vaclav Kotesovec, Dec 18 2019

A361148 a(n) = phi(n)^4.

Original entry on oeis.org

1, 1, 16, 16, 256, 16, 1296, 256, 1296, 256, 10000, 256, 20736, 1296, 4096, 4096, 65536, 1296, 104976, 4096, 20736, 10000, 234256, 4096, 160000, 20736, 104976, 20736, 614656, 4096, 810000, 65536, 160000, 65536, 331776, 20736, 1679616, 104976, 331776, 65536, 2560000

Offset: 1

Vaclav Kotesovec, Mar 02 2023

In general, for k>=1, Sum_{m=1..n} phi(m)^k ~ c(k) * n^(k+1) / (k+1).
Table of the first twenty constants c(k):
c1  = 0.6079271018540266286632767792583658334261526480334792930736...
c2  = 0.4282495056770944402187657075818235461212985133559361440319...
c3  = 0.3371878737915899719616928161521582449491541277581639388802...
c4  = 0.2862564715115608911732883400866386479560747005250468681615...
c5  = 0.2550316684059564308661179534476184539887434047229867871927...
c6  = 0.2342690874743831026992085481001750961630443094403694748409...
c7  = 0.2194845388428573186801010214226853865762414525869501954550...
c8  = 0.2083553180392308846240883587603960475166426933863125773262...
c9  = 0.1996016550942289223053750541784521301740825495040856984950...
c10 = 0.1924764951305819663569723926235916851341834741671794581256...
c11 = 0.1865198318046079731059147989571847359151227252097897755685...
c12 = 0.1814343147960482243026212589426877406632573154701351352790...
c13 = 0.1770192204728143035012153190352692532613146649385520287635...
c14 = 0.1731338036872585521607716180505314246174563305338731073703...
c15 = 0.1696760784770144194638735708052066949428247152918280392147...
c16 = 0.1665700322333281768929516390245288052095235102037486400080...
c17 = 0.1637576294807392765019551841269187995536332906534705685240...
c18 = 0.1611936368897236567526886186599877745065426644021588804182...
c19 = 0.1588421683609925408830108209202958349394621277940566066627...
c20 = 0.1566743130878534775247182243921577941535243896576096188342...
c1 = A059956 = 6/Pi^2, c2 = A065464.
Conjecture: c(k)*log(k) converges to a constant (around 0.534).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of c(k)*log(k), for k = 1..350

Cf. A000010, A002088, A127473, A057434, A358714, A361132, A361179.

Cf. A059956, A065464.

Mathematica
```
Table[EulerPhi[n]^4, {n, 1, 50}]
```
PARI
```
a(n) = eulerphi(n)^4;
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X - 4*p*X + 6*p^2*X - 4*p^3*X) / (1 - p^4*X))[n], ", "))

Multiplicative with a(p^e) = (p-1)^4 * p^(4*e-4).
Dirichlet g.f.: zeta(s-4) * Product_{primes p} (1 + 1/p^s - 4/p^(s-1) + 6/p^(s-2) - 4/p^(s-3)).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = Product_{primes p} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.286256471511560891173288340086638647956074700525046868161...
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^4/((p-1)^4*(p^4-1))) = 2.20815077889083518654... . - Amiram Eldar, Sep 01 2023

A109695 Decimal expansion of Sum_{n>=1} 1/phi(n)^2.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Aug 07 2005

The logarithm of the value can be expanded in a series Sum_{j>=2} c(j)*P(j) = P(2) + 2*P(3) + (7/2)*P(4) + ... where P(.) is the prime zeta function. The partial sums of the series are a slowly oscillating function of the upper limit of j, from which the bracketing interval [3.390642005572503655..., 3.390642005572504756...] for the constant can be computed. - R. J. Mathar, Feb 03 2009

Sum_{n>=1} 1/phi(n)^k is convergent iff k > 1 (reference Monier). - Bernard Schott, Dec 13 2020

			3.39064200557250391614259566300263079374053738121447169118...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21, pp. 281 and 294.

Cf. A000010, A065484, A127473, A335818.

$MaxExtraPrecision = 1000; f[p_] := (1 + p^2/((p - 1)^2*(p^2 - 1))); Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 25 2020 *)

my(N=1000000000); prodeuler(p=2,N,1.+p^2/((p-1)^2*(p^2-1)))*(1+1/(N*log(N)))

prodeulerrat(1 + p^2/((p-1)^2*(p^2-1))) \\ Amiram Eldar, Mar 15 2021

Equals Product_p Sum_{k>=0} 1/phi(p^k)^2 = Product_p (1 + p^2/((p-1)^2*(p^2-1))).

Equals Sum{n>=1} 1/A127473(n). - Amiram Eldar, Mar 15 2021

Four more digits from R. J. Mathar, Feb 03 2009, 25 more Dec 18 2010

More digits from Vaclav Kotesovec, Jun 25 2020

A356533 a(n) = sigma_2(n)^2.

Original entry on oeis.org

1, 25, 100, 441, 676, 2500, 2500, 7225, 8281, 16900, 14884, 44100, 28900, 62500, 67600, 116281, 84100, 207025, 131044, 298116, 250000, 372100, 280900, 722500, 423801, 722500, 672400, 1102500, 708964, 1690000, 925444, 1863225, 1488400, 2102500, 1690000, 3651921

Offset: 1

Vaclav Kotesovec, Aug 11 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (15), a=b=2.

Cf. A001157, A127473, A035116, A072861, A356535 (partial sums).

Mathematica
```
Table[DivisorSigma[2, n]^2, {n, 1, 40}]
```

a(n) = sigma(n, 2)^2; \\ Michel Marcus, Aug 11 2022

Dirichlet g.f.: zeta(s) * zeta(s-2)^2 * zeta(s-4) / zeta(2*s-4).
Multiplicative with a(p^e) = ((p^(2*e+2)-1)/(p^2-1))^2. - Amiram Eldar, Aug 11 2022
a(n) = A001157(n)^2. - R. J. Mathar, Aug 18 2022

A356534 a(n) = sigma_3(n)^2.

Original entry on oeis.org

1, 81, 784, 5329, 15876, 63504, 118336, 342225, 573049, 1285956, 1774224, 4177936, 4831204, 9585216, 12446784, 21911761, 24147396, 46416969, 47059600, 84603204, 92775424, 143712144, 148060224, 268304400, 248094001, 391327524, 417793600, 630612544, 594872100

Offset: 1

Vaclav Kotesovec, Aug 11 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (15), a=b=3.

Cf. A001158, A127473, A035116, A072861, A356536 (partial sums).

Mathematica
```
Table[DivisorSigma[3, n]^2, {n, 1, 40}]
```

a(n) = sigma(n, 3)^2; \\ Michel Marcus, Aug 11 2022

Dirichlet g.f.: zeta(s) * zeta(s-3)^2 * zeta(s-6) / zeta(2*s-6).

Multiplicative with a(p^e) = ((p^(3*e+3)-1)/(p^3-1))^2. - Amiram Eldar, Aug 11 2022

A077101 a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

Original entry on oeis.org

0, 8, 12, 45, 20, 140, 28, 209, 133, 308, 44, 768, 52, 540, 512, 897, 68, 1485, 76, 1700, 880, 1196, 92, 3536, 561, 1620, 1276, 2992, 116, 5120, 124, 3713, 1904, 2660, 1728, 8137, 148, 3276, 2560, 7844, 164, 9072, 172, 6656, 5508, 4700, 188, 15120, 1485

Offset: 1

Labos Elemer, Nov 06 2002

If n is prime, then a(n) = 4n.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A002117, A051612, A062354, A065387, A065464, A072861, A077099, A077100, A127473.

Table[DivisorSigma[1,n]^2-EulerPhi[n]^2,{n,50}] (* Harvey P. Dale, Nov 08 2013 *)

A077101(n) = (sigma(n)^2 - eulerphi(n)^2); \\ Antti Karttunen, May 26 2017

a(n) = A077099(n) * A077100(n). - Antti Karttunen, May 26 2017
From Amiram Eldar, Dec 04 2023: (Start)
a(n) = A072861(n) - A127473(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 5*zeta(3)/2 - Product_{p prime}(1 - (2*p-1)/p^3) = (5/2)*A002117 - A065464 = 2.576892... . (End)

Edited by Dean Hickerson, Nov 07 2002

A082953 a(n) = A000252(n) / A070732(n).

Original entry on oeis.org

1, 2, 4, 8, 16, 8, 36, 32, 36, 32, 100, 32, 144, 72, 64, 128, 256, 72, 324, 128, 144, 200, 484, 128, 400, 288, 324, 288, 784, 128, 900, 512, 400, 512, 576, 288, 1296, 648, 576, 512, 1600, 288, 1764, 800, 576, 968, 2116

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003

From Jianing Song, Apr 20 2019: (Start)
a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].
a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)

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

Cf. A000252, A065464, A070732.
Cf. A000010, A062570, A040001.
Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).

A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
seq(A082953(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

Array[Times @@ Map[EulerPhi, {#, 2 #}] &, 47] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = eulerphi(n)*eulerphi(2*n); \\ Michel Marcus, Jun 04 2025

a(n) = phi(n)*phi(2*n) = A000010(n)*A062570(n). - Vladeta Jovovic, May 02 2005
Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - R. J. Mathar, Apr 14 2011
a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - Jianing Song, Apr 20 2019
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 =  0.171299... . - Amiram Eldar, Oct 30 2022
a(n) = gcd(n,2)*phi(n)^2 = A040001(n)*A127473(n). - Ridouane Oudra, Jun 04 2025

A356535 a(n) = Sum_{k=1..n} sigma_2(k)^2.

Original entry on oeis.org

1, 26, 126, 567, 1243, 3743, 6243, 13468, 21749, 38649, 53533, 97633, 126533, 189033, 256633, 372914, 457014, 664039, 795083, 1093199, 1343199, 1715299, 1996199, 2718699, 3142500, 3865000, 4537400, 5639900, 6348864, 8038864, 8964308, 10827533, 12315933, 14418433

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356533.

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

Cf. A001157, A057434, A061502, A072379, A356536.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[2, k]^2, {k, 1, n}], {n, 1, 40}]

a(n) = sum(k=1, n, sigma(k, 2)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ 189 * zeta(3)^2 * zeta(5) * n^5 / Pi^6.

A356536 a(n) = Sum_{k=1..n} sigma_3(k)^2.

Original entry on oeis.org

1, 82, 866, 6195, 22071, 85575, 203911, 546136, 1119185, 2405141, 4179365, 8357301, 13188505, 22773721, 35220505, 57132266, 81279662, 127696631, 174756231, 259359435, 352134859, 495847003, 643907227, 912211627, 1160305628, 1551633152, 1969426752, 2600039296

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356534.

In general, for m>0, Sum_{k=1..n} sigma_m(k)^2 ~ zeta(2*m+1) * zeta(m+1)^2 * n^(2*m+1) / ((2*m+1) * zeta(2*m+2)).

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

Cf. A001158, A057434, A061502, A072379, A356535.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[3, k]^2, {k, 1, n}], {n, 1, 40}]
Accumulate[DivisorSigma[3,Range[40]]^2] (* This program is much more efficient than the first program above. *) (* Harvey P. Dale, Feb 27 2023 *)

a(n) = sum(k=1, n, sigma(k, 3)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ zeta(7) * n^7 / 6.

cp's OEIS Frontend

A029939 a(n) = Sum_{d|n} phi(d)^2.

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A057434 a(n) = Sum_{k=1..n} phi(k)^2.

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A361148 a(n) = phi(n)^4.

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A109695 Decimal expansion of Sum_{n>=1} 1/phi(n)^2.

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A356533 a(n) = sigma_2(n)^2.

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A356534 a(n) = sigma_3(n)^2.

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A077101 a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

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