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

1,2

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

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))).

Mathematica

f[p_, e_] := 1 + DivisorSum[e, p^# &, CoprimeQ[#, e/#] &]; 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(gcd(d, f[i, 2]/d) == 1, f[i, 1]^d))); }

Crossrefs

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

Sequence in context: A366903 A049418 A383867 * A383866 A333926 A051378

Adjacent sequences: A383861 A383862 A383863 * A383865 A383866 A383867

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 12 2025

Status

approved