A390807 The sum of the integers k from 1 to n such that gcd(n, k) is cubefree (A004709).

1, 3, 6, 10, 15, 21, 28, 28, 45, 55, 66, 78, 91, 105, 120, 112, 153, 171, 190, 210, 231, 253, 276, 252, 325, 351, 351, 406, 435, 465, 496, 448, 561, 595, 630, 666, 703, 741, 780, 700, 861, 903, 946, 990, 1035, 1081, 1128, 1008, 1225, 1275, 1326, 1378, 1431, 1404

Offset

1,2

Comments

The number of these integers is A254926(n).

Links

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

Formula

a(n) = Sum_{k=1..n, gcd(k,n) is cubefree} k = Sum_{k=1..n} A212793(gcd(k,n)) * k.

a(n) = (n + A212793(n)) * A254926(n) / 2.

a(n) <= n*(n+1)/2, with equality if and only if n is cubefree.

a(n) >= A390806(n), with equality if and only if n is squarefree (A005117).

Dirichlet g.f.: (zeta(s-2) + zeta(s-1))/(2 * zeta(3*s-3)).

Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(6)).

Mathematica

f[p_, e_] := If[e < 3, p^e, p^e - p^(e - 3)]; a[n_] := Module[{fct = FactorInteger[n]}, (n + Boole[Max[fct[[;; , 2]]] < 3]) * Times @@ f @@@ fct]/2; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); (n + if(n == 1, 1, vecmax(f[, 2]) < 3)) * prod(i = 1, #f~, f[i, 1]^f[i, 2] - if(f[i, 2] > 2,  f[i, 1]^(f[i, 2]-3))) / 2; }

Crossrefs

Cf. A004709, A005117, A013664, A212793, A254926.

The sum of the integers k from 1 to n such that gcd(n, k) is: A023896 (1), A119790 (prime power, A246655), A390800 (power of prime, A000961), A390801 (prime), A390802 (odd), A390803 (5-rough), A390804 (power of 2), A390805 (3-smooth), A390806 (squarefree), this sequence (cubefree), A390808 (square), A390809 (1 or 2).

Sequence in context: A249736 A371473 A130486 * A054636 A231680 A382726

Adjacent sequences: A390804 A390805 A390806 * A390808 A390809 A390810

Keyword

nonn,easy

Author

Amiram Eldar, Nov 20 2025

Status

approved