A345261 a(n) = Sum_{d|n} d * rad(d).

1, 5, 10, 13, 26, 50, 50, 29, 37, 130, 122, 130, 170, 250, 260, 61, 290, 185, 362, 338, 500, 610, 530, 290, 151, 850, 118, 650, 842, 1300, 962, 125, 1220, 1450, 1300, 481, 1370, 1810, 1700, 754, 1682, 2500, 1850, 1586, 962, 2650, 2210, 610, 393, 755, 2900, 2210, 2810, 590

Offset

1,2

Comments

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

Inverse Möbius transform of n * rad(n). - Wesley Ivan Hurt, Mar 31 2025

Links

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

Formula

a(prime(n)) = A066872(n). - Michel Marcus, Jun 12 2021

From Amiram Eldar, Oct 30 2025: (Start)

Multiplicative with a(p^e) = (p^(e+2) - 1)/(p - 1) - p.

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = A013661 * A002117 * A065464 = 0.84677961058798544766... . (End)

Example

a(10) = Sum_{d|10} d * rad(d) = 1*1 + 2*2 + 5*5 + 10*10 = 1 + 4 + 25 + 100 = 130.

Maple

f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(1 + t[1]^2*(t[1]^t[2]-1)/(t[1]-1), t = F)
end proc:
map(f, [$1..100]); # Robert Israel, Oct 30 2025

Mathematica

Table[Sum[i (1 - Ceiling[n/i] + Floor[n/i]) Product[k^((PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[i/k] + Floor[i/k])), {k, i}], {i, n}], {n, 80}]
f[p_, e_] := (p^(e + 2) - 1)/(p - 1) - p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2025 *)

Prog

(PARI) a(n) = sumdiv(n, d, d*factorback(factorint(d)[, 1])); \\ Michel Marcus, Oct 30 2025
(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(e+2) - 1)/(p - 1) - p); } \\ Amiram Eldar, Oct 30 2025

Crossrefs

Cf. A007947 (rad), A066872, A345263.

Cf. A002117, A013661, A065464.

Sequence in context: A272267 A195735 A061145 * A321558 A328667 A317966

Adjacent sequences: A345258 A345259 A345260 * A345262 A345263 A345264

Keyword

nonn,mult,easy

Author

Wesley Ivan Hurt, Jun 12 2021

Status

approved