A363334 a(n) is the sum of divisors of n that are both coreful and bi-unitary.

1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 26, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 62, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 78, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 118, 65, 66, 67

Offset

1,2

Comments

First differs from A363331 at n = 16.

The number of these divisors is A363332(n).

Links

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

Formula

Multiplicative with a(p^e) = (p^(e+1) - 1)/(p - 1) - 1, if e is odd, and (p^(e+1) - 1)/(p - 1) - p^(e/2) - 1 if e is even.

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

a(n) >= A362852(n), with equality if and only if n = 1.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (p/(p+1))*(1+1/p-1/p^3+2/p^5) = 0.557782322450569540209... .

Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(3*s-2) + 2/p^(3*s-1) - 2/p^(4*s-2)). - Amiram Eldar, Oct 01 2023

Example

a(8) = 14 since 8 has 3 divisors that are both bi-unitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.

Mathematica

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - 1 - If[OddQ[e], 0, p^(e/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~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1 - if(f[i, 2]%2, 0, f[i, 1]^(f[i, 2]/2))); }

Crossrefs

Cf. A004709, A057723, A188999, A222266, A362852, A363331, A363332.

Sequence in context: A319723 A271377 A363331 * A066336 A076133 A277908

Adjacent sequences: A363331 A363332 A363333 * A363335 A363336 A363337

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 28 2023

Status

approved