A384555 The largest infinitary divisor of n that is cubefree.

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset

1,2

Comments

The number of these divisors is A368883(n), and their sum is A384554(n).

Links

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

Formula

Multiplicative with a(p^e) = 1 if e == 0 (mod 4), p if e == 1 (mod 4), p^2 if e == 2 or 3 (mod 4).

a(n) = n if and only if n is cubefree (A004709).

Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(3*s-2)).

Sum_{k=1..n} a(k) = c * n^2 / 2, where c = zeta(8) * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 1/p^5) = 0.87406992849637563411... .

Mathematica

f[p_, e_] := Switch[Mod[e, 4], 0, 1, 1, p, 2, p^2, 3, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; [1, p, p^2, p^2][e%4+1]); }

Crossrefs

Cf. A004709, A007948, A013666, A077609, A368883, A384554.

Sequence in context: A319999 A083501 A007922 * A007948 A377518 A355261

Adjacent sequences: A384552 A384553 A384554 * A384556 A384557 A384558

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 03 2025

Status

approved