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

Author

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

Keywords

• nonn
• mult

Comments

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).

Links

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

Crossrefs

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

Cf. A082695 (c1).

Programs

Mathematica

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

PARI

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], ", "))

Formula

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...