A361266 Multiplicative with a(p^e) = p^(e + 3), e > 0.

1, 16, 81, 32, 625, 1296, 2401, 64, 243, 10000, 14641, 2592, 28561, 38416, 50625, 128, 83521, 3888, 130321, 20000, 194481, 234256, 279841, 5184, 3125, 456976, 729, 76832, 707281, 810000, 923521, 256, 1185921, 1336336, 1500625, 7776, 1874161, 2085136, 2313441, 40000

Offset: 1

Author

Vaclav Kotesovec, Mar 06 2023

Keywords

• nonn
• easy
• mult

Comments

In general, if the function is multiplicative with a(p^e) = p^(e + m) where m > 0, then Dirichlet g.f.: Product_{primes p} (1 + p^(m+1)/(p^s - p)).

Equivalently, Dirichlet g.f.: zeta(s-m-1) * zeta(s-1) * Product_{primes p} (1 + p^(2 + m - 2*s) - p^(2 + 2*m - 2*s) - p^(1 - s)).

Sum_{k=1..n} a(k) ~ c(m) * zeta(m+1) * n^(m+2) / (m+2), where c(m) = Product_{primes p} (1 - 1/p^2 - 1/p^(m+1) + 1/p^(m+2)).

Limit_{m->oo} c(m) = 6/Pi^2 = A059956.

Links

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

Crossrefs

Cf. A003557, A007947, A064549, A361264.

Programs

Maple

f:= proc(n) local t;
  mul(t[1]^(t[2]+3), t = ifactors(n)[2])
end proc:
map(f, [$1..50]); # Robert Israel, Mar 07 2023

Mathematica

g[p_, e_] := p^(e+3); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

PARI

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

PARI

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,2] +=3); factorback(f); \\ Michel Marcus, Mar 07 2023

Formula

Dirichlet g.f.: Product_{primes p} (1 + p^4/(p^s - p)).

Dirichlet g.f.: zeta(s-4) * zeta(s-1) * Product_{primes p} (1 + p^(5 - 2*s) - p^(8 - 2*s) - p^(1 - s)).

Sum_{k=1..n} a(k) ~ c * Pi^4 * n^5 / 450, where c = Product_{primes p} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.5761527353856670595206110782641172754062471168028961885...

From Amiram Eldar, Sep 01 2023: (Start)

a(n) = n * A007947(n)^3 = A064549(n) * A007947(n)^2 = A361264(n) * A007947(n) = A064549(A064549(A064549(n))).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.148846213921... . (End)