A368170 -id:A368170 - cp's OEIS frontend

A368172 The number of divisors of the largest cubefull exponentially odd divisor of n (A368170).

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A365487 at n = 32.

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

Cf. A000005, A013661, A004709, A286324, A365487, A368170.

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

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

a(n) = A000005(A368170(n)).
Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = e+1 if e is odd and e > 1, and a(p^e) = e otherwise.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A000005(n), with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 - 1/p^5) = 1.69824776889117043774... .

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

Original entry on oeis.org

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

Amiram Eldar, Dec 14 2023

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

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

Cf. A004709, A335988, A368170.

Cf. A050985, A367699.

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]

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])))};

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... .

cp's OEIS Frontend

A368172 The number of divisors of the largest cubefull exponentially odd divisor of n (A368170).

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