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

1,2

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

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... .

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]

Prog

(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])))};

Crossrefs

Cf. A004709, A335988, A368170.

Cf. A050985, A367699.

Sequence in context: A360539 A319656 A367699 * A050985 A367168 A367514

Adjacent sequences: A368168 A368169 A368170 * A368172 A368173 A368174

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 14 2023

Status

approved