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

1,2

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

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))).

Mathematica

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

Prog

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

Crossrefs

Cf. A051378, A209061, A322791, A361174, A383761, A383863.

Sequence in context: A353900 A366903 A049418 * A383864 A383866 A333926

Adjacent sequences: A383864 A383865 A383866 * A383868 A383869 A383870

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 13 2025

Status

approved