A377517 The sum of the divisors of n that are terms in A276078.

1, 3, 4, 3, 6, 12, 8, 3, 13, 18, 12, 12, 14, 24, 24, 3, 18, 39, 20, 18, 32, 36, 24, 12, 31, 42, 13, 24, 30, 72, 32, 3, 48, 54, 48, 39, 38, 60, 56, 18, 42, 96, 44, 36, 78, 72, 48, 12, 57, 93, 72, 42, 54, 39, 72, 24, 80, 90, 60, 72, 62, 96, 104, 3, 84, 144, 68, 54

Offset: 1

Author

Amiram Eldar, Oct 30 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A046897 at n = 27 = 3^3: a(27) = 13, while A046897(27) = 40.

The number of these divisors is A377516(n), and the largest of them is A377515(n).

Links

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

Crossrefs

Cf. A000203, A000720, A013661, A046897, A276078, A377515, A377516, A377520.

Programs

Mathematica

f[p_, e_] := (p^(Min[PrimePi[p], e] + 1) - 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]^(min(primepi(f[i,1]), f[i,2]) + 1) - 1)/(f[i,1] - 1));}

Formula

a(n) = A000203(A377515(n)).

Multiplicative with a(p^e) = (p^(min(pi(p), e)+1) - 1)/(p - 1), where pi(n) = A000720(n).

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^(pi(p)+1)) = 1.18603586369737251334... .