A345264 a(n) = Sum_{d|n} rad(d) * mu(n/d)^2.

1, 3, 4, 4, 6, 12, 8, 4, 6, 18, 12, 16, 14, 24, 24, 4, 18, 18, 20, 24, 32, 36, 24, 16, 10, 42, 6, 32, 30, 72, 32, 4, 48, 54, 48, 24, 38, 60, 56, 24, 42, 96, 44, 48, 36, 72, 48, 16, 14, 30, 72, 56, 54, 18, 72, 32, 80, 90, 60, 96, 62, 96, 48, 4, 84, 144, 68, 72, 96, 144, 72, 24

Offset

1,2

Comments

If p is prime, a(p) = Sum_{d|p} rad(d) * mu(p/d)^2 = 1*1 + p*1 = p + 1.

Links

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

Formula

From Amiram Eldar, Oct 30 2025: (Start)

Multiplicative with a(p) = p + 1, and a(p^e) = 2*p for e >= 2.

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 2/p^4 + 1/p^5) = 1.07062376419273610054... . (End)

Example

a(8) = Sum_{d|8} rad(d) * mu(8/d)^2 = 1*0 + 2*0 + 2*1 + 2*1 = 4.

Maple

f:= proc(n) local F, F1, F2, w, t, S, J, J1, i;
  F:= ifactors(n)[2]; w:= nops(F);
  F1:= F[.., 1]; F2:= F[.., 2]; J1:= select(i -> F2[i]>1, {$1..w});
  t:= 0;
  for S in combinat:-powerset({$1..w}) do
    J:= S union J1;
    t:= t + mul(F1[i], i=J);
  od;
  t;
end proc:
map(f, [$1..100]); # Robert Israel, Oct 30 2025

Mathematica

Table[Sum[(1 - Ceiling[n/i] + Floor[n/i]) MoebiusMu[n/i]^2 Product[k^((PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[i/k] + Floor[i/k])), {k, i}], {i, n}], {n, 100}]
f[p_, e_] := If[e == 1, p+1, 2*p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2025 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1]+1, 2*f[i, 1])); } \\ Amiram Eldar, Oct 30 2025

Crossrefs

Cf. A007947 (rad).

Sequence in context: A089640 A086659 A265887 * A344465 A008473 A326043

Adjacent sequences: A345261 A345262 A345263 * A345265 A345266 A345267

Keyword

nonn,mult,easy

Author

Wesley Ivan Hurt, Jun 12 2021

Status

approved