A361148 -id:A361148 - cp's OEIS frontend

A127473 a(n) = phi(n)^2.

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 64, 64, 256, 36, 324, 64, 144, 100, 484, 64, 400, 144, 324, 144, 784, 64, 900, 256, 400, 256, 576, 144, 1296, 324, 576, 256, 1600, 144, 1764, 400, 576, 484, 2116, 256, 1764, 400, 1024, 576, 2704, 324, 1600

Offset: 1

Gary W. Adamson, Jan 15 2007

Number of maps of the form j |--> m*j + d with gcd(m, n) = 1 and gcd(d, n) = 1 from [1, 2, ..., n] to itself. - Joerg Arndt, Aug 29 2014
Right border of A127474.
Equals the Mobius transform (A054525) of A029939. - Gary W. Adamson, Aug 20 2008
From Jianing Song, Apr 14 2019: (Start)
a(n) is the number of solutions to gcd(xy, n) = 1 with x, y in [0, n-1].
Let Z_n be the ring of integers modulo n, then a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - x) (or equivalently, Z_n[x]/(x^2 + x)) with discriminant d = 1 (that is, a(n) is the size of the group G(n) = (Z_n[x]/(x^2 - x))*). Actually, G(n) is isomorphic to (Z_n)* X (Z_n)*. (End)

			a(5) = 16 since phi(5) = 4.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Cf. A000010, A057434, A065464, A109695, A127474, A358714, A361148, A008683.

Similar sequences: A082953 (size of (Z_n[x]/(x^2 - 1))*, d = 4), 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).

[(EulerPhi(n))^2: n in [1..180]]; // Vincenzo Librandi, Apr 04 2011

A127473 := proc(n) numtheory[phi](n)^2 ; end proc:
seq(A127473(n),n=1..40) ; # R. J. Mathar, Apr 04 2011

Table[EulerPhi[n]^2,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

a(n) = eulerphi(n)^2; \\ Michel Marcus, Oct 16 2018

a(n) = A000010(n)^2.
Multiplicative with a(p^e) = (p-1)^2*p^(2e-2), e >= 1. Dirichlet g.f. zeta(s-2)*Product_{primes p} (1 - 2/p^(s-1) + 1/p^s). - R. J. Mathar, Apr 04 2011
Sum_{k>=1} 1/a(k) = A109695. - Vaclav Kotesovec, Sep 20 2020
Sum_{k>=1} (-1)^k/a(k) = (1/7) * A109695. - Amiram Eldar, Nov 11 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime}(1 - (2*p-1)/p^3) = A065464 / 3 = 0.142749... . - Amiram Eldar, Oct 25 2022
a(n) = Sum_{d|n} mu(n/d)*phi(n*d). - Ridouane Oudra, Jul 23 2025

A361132 Multiplicative with a(p^e) = e^4, p prime and e > 0.

Original entry on oeis.org

1, 1, 1, 16, 1, 1, 1, 81, 16, 1, 1, 16, 1, 1, 1, 256, 1, 16, 1, 16, 1, 1, 1, 81, 16, 1, 81, 16, 1, 1, 1, 625, 1, 1, 1, 256, 1, 1, 1, 81, 1, 1, 1, 16, 16, 1, 1, 256, 16, 16, 1, 16, 1, 81, 1, 81, 1, 1, 1, 16, 1, 1, 16, 1296, 1, 1, 1, 16, 1, 1, 1, 1296, 1, 1, 16, 16

Offset: 1

Vaclav Kotesovec, Mar 02 2023, following a suggestion from Amiram Eldar

In general, if the function is multiplicative with a(p^e) = e^k, where k>=1, then Sum_{m=1..n} a(m) ~ c(k) * n, where c(k) = Product_{primes p} (1 + PolyLog(-k, 1/p)) * (1 - 1/p).
Equivalently, c(k) = Product_{primes p} (1 + Sum_{j>=2} (j^k - (j-1)^k) / p^j).
Sum_{m=1..n} A005361(m)^k ~ c(k) * n.
Table of logarithms of the first twenty constants c(k):
log(c1)  =    0.6645400902595784780106197346845697376257107319484837534113838...
log(c2)  =    2.1027190979191945200514651557327047986978773488049101019457040...
log(c3)  =    4.6968549904993458045898305766669061238379561861949323835425304...
log(c4)  =    8.6865032221694100694964858752580123427478996289429265630701524...
log(c5)  =   14.2913129298819954890384122051888143114132125173972994127345117...
log(c6)  =   21.8135511355940060754244319875442802379763506456537810297977335...
log(c7)  =   31.6936244245134941047326145621097555406387768809071583785926496...
log(c8)  =   44.5357450879229051636129496942971942282070021854681649075237793...
log(c9)  =   61.1279313139359633940353674601273793850149492879803908371116076...
log(c10) =   82.5520903493060704390063479960346732401820956158379186266389560...
log(c11) =  110.2954981238150788264027780431082219466660734768697563026966486...
log(c12) =  146.3390378386537094475359791093275236623437203145309460650602987...
log(c13) =  193.3102629498150337396691694808577709247583271151043344733643302...
log(c14) =  254.7562108044458078036208253682699240853829328072028848109791635...
log(c15) =  335.5155584889434205169760027607421364026263435517505529418223175...
log(c16) =  442.1708823748701851244490135727342670822854621013078138839028927...
log(c17) =  583.6971600757633563987486782501478518757572163549653222049269791...
log(c18) =  772.3363960260522276224001927946529683262139600086441840227950538...
log(c19) = 1024.7789861796186438478485897805332932014500908873437888887485298...
log(c20) = 1363.8429394936892771815120584792965902670785987496833459129791344...
Conjecture: log(log(c(k)))/k converges to a constant (around 0.315).

Vaclav Kotesovec, Plot of log(log(c(k))) / k, for k = 1..40

Cf. A005361, A360969, A360970, A322328, A361148, A361179.

Cf. A082695 (c1).

g[p_, e_] := e^4; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1 - 4*X + 21*X^2 + X^3 + 6*X^4 - X^5)/(1-X)^5)[n], ", "))

a(n) = A005361(n)^4.
Dirichlet g.f.: Product_{primes p} (1 + p^s*(p^(3*s) + 11*p^(2*s) + 11*p^s + 1) / (p^s - 1)^5).
Sum_{k=1..n} a(k) ~ c * n, where c = Product_{primes p} (1 + (15*p^3 + 5*p^2 + 5*p - 1) / (p*(p-1)^4)) = 5922.43654748315227690838901234893132297258444672...

A361179 a(n) = sigma(n)^4.

Original entry on oeis.org

1, 81, 256, 2401, 1296, 20736, 4096, 50625, 28561, 104976, 20736, 614656, 38416, 331776, 331776, 923521, 104976, 2313441, 160000, 3111696, 1048576, 1679616, 331776, 12960000, 923521, 3111696, 2560000, 9834496, 810000, 26873856, 1048576, 15752961, 5308416

Offset: 1

Vaclav Kotesovec, Mar 03 2023

In general, for k>=1, Sum_{m=1..n} sigma(m)^k ~ c(k) * z(k) * n^(k+1) / (k+1), where z(k) = Product_{j=2..k+1} zeta(j).
z(k) tends to A021002 = 2.29485659167331379418351583... if k tends to infinity.
Table of logarithms of the first twenty constants c(k):
log(c1)  = 0
log(c2)  = 0.4185904294034097177091498674425959208785022862606440306200960821...
log(c3)  = 1.0423888168104400391462790418324165821902123159643681963298587386...
log(c4)  = 1.7991790110714031081639242851527957388041981665455193670488985855...
log(c5)  = 2.6531418047626712704435945717713008165192112256395129469527055461...
log(c6)  = 3.5826667694785981489341382260447390026333883927530294731356708082...
log(c7)  = 4.5733843557245275039380976990636718508529417039225677910093512418...
log(c8)  = 5.6152065176325962438798772352645945078887296036246579568363264836...
log(c9)  = 6.7007695219862872061684609152917692899880931107656334442026270254...
log(c10) = 7.8245175718301572361518558972457980392624870372412384620464547480...
log(c11) = 8.9821318589248960303876549202030018215854310738197659104984082438...
log(c12) = 10.170161510396427442300796140752106239603402200741405656518889304...
log(c13) = 11.385778844373902103940190311048453116470874526205115584130363228...
log(c14) = 12.626614423444098003503814842580453502016287945932183786430620101...
log(c15) = 13.890644760144907314506933347339629337810929043024214330654043796...
log(c16) = 15.176115136560648867246990011975416479066956527530401883224856531...
log(c17) = 16.481485806132270823150284520463000397265757050340939883069076823...
log(c18) = 17.805393674783928883671133007206209125657866860089528876021281793...
log(c19) = 19.146624201995507049618714377273936711664382470319966849198205155...
log(c20) = 20.504090088752226662590920186246482636058069128320785639131816842...
c1 = 1, c2 = 5/(2*zeta(2)) = 15/Pi^2.

Cf. A000203, A072861, A361147.

Cf. A361132, A361148.

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

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

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^4.
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * zeta(s-4) * Product_{primes p} (1 + 1/p^(3*s-6) + 3/p^(2*s-3) + 5/p^(2*s-4) + 3/p^(2*s-5) + 3/p^(s-1) + 5/p^(s-2) + 3/p^(s-3)).
Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * zeta(5) * n^5 / 2700, where c = Product_{primes p} (1 + 3/p^2 + 5/p^3 + 3/p^4 + 3/p^5 + 5/p^6 + 3/p^7 + 1/p^9) = 6.0446828090651437986928739783339791032197283386377841627594461874871547391...
a(n) = A000583(A000203(n)).

A358714 a(n) = phi(n)^3.

Original entry on oeis.org

1, 1, 8, 8, 64, 8, 216, 64, 216, 64, 1000, 64, 1728, 216, 512, 512, 4096, 216, 5832, 512, 1728, 1000, 10648, 512, 8000, 1728, 5832, 1728, 21952, 512, 27000, 4096, 8000, 4096, 13824, 1728, 46656, 5832, 13824, 4096, 64000, 1728, 74088, 8000, 13824, 10648, 97336, 4096, 74088

Offset: 1

Param Mayurkumar Parekh and Paavan Mayurkumar Parekh, Nov 28 2022

Number of solutions to gcd(x*y*z, n) = 1 such that 0 <= x,y,z <= n-1.

x*y*z == t (mod n) where t is a unit (invertible element) in Z_n. Since t is a unit, all x,y,z must be units. Here there are A000010(n) possibilities for each x,y,z so there are a total of A000010(n)^3 ways to get t as a unit.

			a(9) = A000010(9)^3 = 216.

Cf. A000010, A127473, A361148, A335818.

Magma
```
[(EulerPhi(n))^3: n in [1..180]];
```

a[n_] := EulerPhi[n]^3; Array[a, 100] (* Amiram Eldar, Jan 06 2023 *)

PARI
```
a(n) = eulerphi(n)^3;
```

a(n) = A000010(n)^3.
From Amiram Eldar, Jan 06 2023: (Start)
Multiplicative with a(p^e) = (p-1)^3*p^(3*e-3).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime}(1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.08429696844... .
Sum_{k>=1} 1/a(k) = Product_{p prime} (1 + p^3/((p-1)^3*(p^3-1))) = 2.47619474816... (A335818). (End)

A377509 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^4 * x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 36, 234, 7290, 54540, 1408680, 23119740, 341788860, 11790437400, 231972879600, 8206299070200, 191673262380600, 6154270418696400, 206515993375692000, 6574758436640394000, 269828090984990538000, 9531096165082736244000, 411037724983993923816000

Offset: 0

Vaclav Kotesovec, Oct 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..412

Cf. A318917, A377507, A377508.

Cf. A205797, A361148.

nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^4 * a[n-k], {k, 1, n}]/n];Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^4 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

log(a(n)/n!) ~ 5 * 3^(1/5) * c^(1/5) * n^(4/5) / 2^(7/5), where c = Product_{primes p} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.286256471511560891173288340086638647956...

cp's OEIS Frontend

A127473 a(n) = phi(n)^2.

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A361132 Multiplicative with a(p^e) = e^4, p prime and e > 0.

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A361179 a(n) = sigma(n)^4.

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A358714 a(n) = phi(n)^3.

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A377509 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^4 * x^k/k), where phi is the Euler totient function A000010.

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