A360969 Multiplicative with a(p^e) = e^2, p prime and e > 0.

1, 1, 1, 4, 1, 1, 1, 9, 4, 1, 1, 4, 1, 1, 1, 16, 1, 4, 1, 4, 1, 1, 1, 9, 4, 1, 9, 4, 1, 1, 1, 25, 1, 1, 1, 16, 1, 1, 1, 9, 1, 1, 1, 4, 4, 1, 1, 16, 4, 4, 1, 4, 1, 9, 1, 9, 1, 1, 1, 4, 1, 1, 4, 36, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 4, 4, 1, 1, 1, 16, 16, 1, 1, 4

Offset: 1

Author

Vaclav Kotesovec, Feb 27 2023

Keywords

• nonn
• mult

Comments

From Bernard Schott, Feb 27 2023: (Start)

The three fixed points are 1, 4 and 16.

a(n) = 1 iff n is A005117.

a(n) = 4 iff n is in A060687. (End)

Links

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

Crossrefs

Cf. A005361, A322327, A360970, A361132.

Cf. A005117, A060687.

Programs

Maple

f:= proc(n) local t;
  mul(t^2, t = ifactors(n)[2][..,2]);
end proc:
map(f, [$1..100]); # Robert Israel, Mar 29 2023

Mathematica

g[p_, e_] := e^2; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

PARI

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

PARI

a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1]=f[k,2]^2; f[k, 2]=1); factorback(f); \\ Michel Marcus, Feb 27 2023

Formula

Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + (3*p^s - 1) / (p^s*(p^s - 1)^2)).

Sum_{k=1..n} a(k) ~ c*n, where c = Product_{primes p} (1 + (3*p - 1) / (p*(p-1)^2)) = 8.18840474382698544967326709964388539461401085196013492328186138...

a(n) = A005361(n)^2.