A361179 -id:A361179 - cp's OEIS frontend

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

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...

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

A205797 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^4 * x^n/n ).

Original entry on oeis.org

1, 1, 41, 126, 1526, 5185, 46920, 176865, 1254608, 4986548, 30563031, 123868761, 683127011, 2793828323, 14223836013, 58127497582, 278433541834, 1130954381904, 5159127957638, 20767403083249, 91032595281699, 362455763000997, 1536849042738162

Offset: 0

Paul D. Hanna, Jan 31 2012

nonn

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

			G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 5185*x^5 +...
such that, by definition,
log(A(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 6^4*x^5/5 + 12^4*x^6/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..3200

Cf. A156302, A178933, A000203 (sigma), A000041 (partitions), A361179.

nmax = 30; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^4 * a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 30 2024 *)

{a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^4*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */

{a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^3*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */

a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^4*a(n-k)))

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^4*a(n-k) for n>0, with a(0) = 1.
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^3 * x^(n*k) / n ).
From Vaclav Kotesovec, Oct 30 2024: (Start)
log(a(n)) ~ 5^(4/5) * c^(1/5) * Pi^(6/5) * zeta(3)^(1/5) * zeta(5)^(1/5) * n^(4/5) / (2^(9/5) * 3^(2/5)), 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) = 6.04468280906514379869287397833397910321972833863778...
Equivalently, log(a(n)) ~ 3.967005157823944635858584839447899089435134... * n^(4/5). (End)

A361147 a(n) = sigma(n)^3.

Original entry on oeis.org

1, 27, 64, 343, 216, 1728, 512, 3375, 2197, 5832, 1728, 21952, 2744, 13824, 13824, 29791, 5832, 59319, 8000, 74088, 32768, 46656, 13824, 216000, 29791, 74088, 64000, 175616, 27000, 373248, 32768, 250047, 110592, 157464, 110592, 753571, 54872, 216000, 175616

Offset: 1

Vaclav Kotesovec, Mar 02 2023

Cf. A000203, A024916, A072861, A072379, A361179.

Cf. A000005, A000578, A035116, A319089.

Mathematica
```
Table[DivisorSigma[1, n]^3, {n, 1, 50}]
```
PARI
```
a(n) = sigma(n)^3;
```

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

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^3.
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * n^4 / 2160, where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.83598357433419286770442457158038489640898183...
a(n) = A000578(A000203(n)).

cp's OEIS Frontend

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

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

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A205797 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^4 * x^n/n ).

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A361147 a(n) = sigma(n)^3.

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