A383867 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 squarefree divisor of the p-adic valuation of n.

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 7, 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, 28, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144, 68

Offset: 1

Author

Amiram Eldar, May 13 2025

Keywords

• nonn
• easy
• mult

Comments

Analogous to the sum of (1+e)-divisors (A051378) as exponential squarefree exponential divisors (A383761, A361174) 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. A051378, A209061, A322791, A361174, A383761, A383863.

Programs

Mathematica

f[p_, e_] := 1 + DivisorSum[e, p^# &, SquareFreeQ[#] &]; 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(issquarefree(d), f[i, 1]^d)));}

Formula

Multiplicative with a(p^e) = 1 + Sum_{d squarefree divisor of e} 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) = 1.47709589136345836345..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, d squarefree} x^(2*k-d))).