A383864 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 19, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 76, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Author

Amiram Eldar, May 12 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A383866 at n = 256.

The sum of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).

Analogous to the sum of (1+e)-divisors (A051378) as exponential unitary divisors (A361255, A322857) are analogous to exponential divisors (A322791, A051377).

The number of these divisors is A383863(n).

Links

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

Crossrefs

Cf. A051377, A209061, A322791, A322857, A361255, A383863, A383866.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sumdiv(f[i, 2], d, if(gcd(d, f[i, 2]/d) == 1, f[i, 1]^d)));}

Formula

Multiplicative with a(p^e) = 1 + Sum_{d|e, gcd(d, e/d) = 1} p^d.

a(n) <= A051378(n), with equality if and only if n is an exponentially squarefree number (A209061).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52168352620962354041..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, gcd(d, k/d)=1} x^(2*k-d))).