A365337 The sum of divisors of the largest exponentially odd number dividing n.

1, 3, 4, 3, 6, 12, 8, 15, 4, 18, 12, 12, 14, 24, 24, 15, 18, 12, 20, 18, 32, 36, 24, 60, 6, 42, 40, 24, 30, 72, 32, 63, 48, 54, 48, 12, 38, 60, 56, 90, 42, 96, 44, 36, 24, 72, 48, 60, 8, 18, 72, 42, 54, 120, 72, 120, 80, 90, 60, 72, 62, 96, 32, 63, 84, 144, 68

Offset: 1

Author

Amiram Eldar, Sep 01 2023

Keywords

• nonn
• easy
• mult

Comments

The number of divisors of the largest exponentially odd number dividing n is A286324(n).

Links

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

Crossrefs

Cf. A000203, A065487, A286324, A350390.

Programs

Mathematica

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

Formula

a(n) = A000203(A350390(n)).

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

Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(3*s-2)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911488886... (A065487). - Amiram Eldar, Sep 01 2023