A391989 The difference between the number of exponential divisors and the number of exponential unitary divisors for numbers that are not exponentially squarefree.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2

Offset

1,6

Comments

The nonzero terms in the sequence {A049419(n) - A278908(n) ; n >= 1}.

For exponentially squarefree numbers the sets of exponential divisors (A322791) and exponential unitary divisors (A361255) are equal.

Links

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

Formula

a(n) = A049419(A130897(n)) - A278908(A130897(n)).

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/(1-d) * (Product_{p prime} (1 + Sum_{k>=2} (tau(k) - tau(k-1))/p^e) - Product_{p prime} (1 + Sum_{k>=2} (2^omega(k) - 2^omega(k-1))/p^k))) = 1.34201066812805721783..., where d = A262276, tau = A000005, and omega = A001221.

Mathematica

s[1] = 0; s[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ DivisorSigma[0, e] - Times @@ (2^PrimeNu[e])]; Select[Array[s, 3000], # > 0 &]

Prog

(PARI) f(n) = {my(e = factor(n)[, 2]); vecprod(apply(x -> numdiv(x), e)) - vecprod(apply(x -> 1 << omega(x), e)); }
list(lim) = select(x -> x > 0, vector(lim, i, f(i)));

Crossrefs

Cf. A000005, A001221, A049419, A130897, A262276, A278908, A322791, A361255, A391988.

Similar sequence: A174961.

Sequence in context: A370080 A372331 A318498 * A093997 A157196 A300410

Adjacent sequences: A391986 A391987 A391988 * A391990 A391991 A391992

Keyword

nonn,easy

Author

Amiram Eldar, Dec 26 2025

Status

approved