A384558 The sum of the exponential divisors of n that are exponentially odd numbers (A268335).

1, 2, 3, 2, 5, 6, 7, 10, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 30, 5, 26, 30, 14, 29, 30, 31, 34, 33, 34, 35, 6, 37, 38, 39, 50, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 60, 55, 70, 57, 58, 59, 30, 61, 62, 21, 10, 65, 66, 67, 34, 69

Offset

1,2

Comments

First differs from A384559 at n = 512: a(512) = 522, while A384559(512) = 514.

The number of these divisors is A368979(n), and the largest of them is A331737(n).

The indices of records of a(n)/n are the primorial numbers (A002110) cubed, i.e., 1 and the terms of A115964.

Links

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

Formula

Multiplicative with a(p^e) = Sum_{d|e, d odd} p^d.

a(n) = n if and only if n is squarefree (A005117).

a(n) < n if and only if n is in A072587.

a(n) > n if and only if n is in A374459.

limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1)) - 1/(p^2-1) + (1-1/p) * Sum_{k>=1} p^(2*k+1)/(p^(4*k+2)-1)) = 0.80824764393216997768... .

Maple

A384558:=proc(n)
    local a, pe, p, e, af, d;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1, pe) ;
        e := op(2, pe) ;
        af := 0 ;
        for d in numtheory[divisors](e) do
            if type(d, 'odd') then
                af := af+p^d ;
            end if;
        end do:
        a := a*af ;
    end do;
    a
end proc:
seq(A384558(n), n=1..100); # R. J. Mathar, Jun 04 2025

Mathematica

f[p_, e_] := DivisorSum[e, p^# &, OddQ[#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i, 2], d, (d % 2) * f[i, 1]^d)); }

Crossrefs

Cf. A002110, A005117, A051377, A072587, A082020, A115964, A268335, A331737, A368979, A374459, A384559.

Sequence in context: A066729 A241592 A211506 * A384559 A182880 A182898

Adjacent sequences: A384555 A384556 A384557 * A384559 A384560 A384561

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 03 2025

Status

approved