A374783 Numerator of the mean unitary abundancy index of the unitary divisors of n.

1, 5, 7, 9, 11, 35, 15, 17, 19, 11, 23, 21, 27, 75, 77, 33, 35, 95, 39, 99, 5, 115, 47, 119, 51, 135, 55, 135, 59, 77, 63, 65, 161, 175, 33, 19, 75, 195, 63, 187, 83, 25, 87, 207, 209, 235, 95, 77, 99, 51, 245, 243, 107, 275, 23, 255, 91, 295, 119, 231, 123, 315

Offset

1,2

Comments

The unitary abundancy index of a number k is A034448(k)/k = A332882(k)/A332883(k).

The record values of a(n)/A374784(n) are attained at the primorial numbers (A002110).

The least number k such that a(k)/A374784(k) is larger than 2, 3, 4, ..., is A002110(9) = 223092870, A002110(314) = 7.488... * 10^878, A002110(65599) = 5.373... * 10^356774, ... .

Links

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

Formula

Let f(n) = a(n)/A374784(n). Then:

f(n) = (Sum_{d|n, gcd(d, n/d) = 1} usigma(d)/d) / ud(n), where usigma(n) is the sum of unitary divisors of n (A034448), and ud(n) is their number (A034444).

f(n) is multiplicative with f(p^e) = 1 + 1/(2*p^e).

f(n) = (Sum_{d|n, gcd(d, n/d) = 1} d*ud(d))/(n*ud(n)) = A343525(n)/(n*A034444(n)).

Dirichlet g.f. of f(n): zeta(s) * zeta(s+1) * Product_{p prime} (1 - 1/(2*p^(s+1)) - 1/(2*p^(2*s+1))).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} (1 + 1/(2*p*(p+1))) = 1.17443669198552182119... . For comparison, the asymptotic mean of the unitary abundancy index over all the positive integers is zeta(2)/zeta(3) = 1.368432... (A306633).

Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).

f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).

Example

For n = 4, 4 has 2 unitary divisors, 1 and 4. Their unitary abundancy indices are usigma(1)/1 = 1 and usigma(4)/4 = 5/4, and their mean unitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = numerator(9/8) = 9.

Mathematica

f[p_, e_] := 1 + 1/(2*p^e); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, 1 + 1/(2*f[i, 1]^f[i, 2]))); }

Crossrefs

Cf. A002110, A005117, A034444, A034448, A077610, A306633, A332882, A332883, A343525, A374784 (denominators).

Similar sequences: A374777/A374778, A374786/A374787.

Sequence in context: A346368 A029664 A188793 * A343525 A374786 A184102

Adjacent sequences: A374780 A374781 A374782 * A374784 A374785 A374786

Keyword

nonn,easy,frac

Author

Amiram Eldar, Jul 20 2024

Status

approved