A368171 a(n) is the smallest divisor d of n such that n/d is a cubefull exponentially odd number (A335988).

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 6, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Amiram Eldar, Dec 14 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A050985 at n = 32, and from A367699 at n = 64.

Links

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

Crossrefs

Cf. A004709, A335988, A368170.

Cf. A050985, A367699.

Programs

Mathematica

f[p_, e_] := If[e <= 2, p^e, If[EvenQ[e], p, 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2] <= 2, f[i,1]^f[i,2], if(f[i,2]%2, 1, f[i,1])))};

Formula

Multiplicative with a(p^e) = p^e if e <= 2, a(p^e) = 1 if e is odd and e > 1, and p otherwise.

a(n) = n/A368170(n).

a(n) >= 1, with equality if and only if n is in A335988.

a(n) <= n, with equality if and only if n is cubefree (A004709).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/30) * Product_{p prime} (1 + 1/p^2 - 1/p^3 - 1/p^5 + 1/p^6) = 0.42246686366220037592... .