A129468 Unitary abundance of n.

-1, -1, -2, -3, -4, 0, -6, -7, -8, -2, -10, -4, -12, -4, -6, -15, -16, -6, -18, -10, -10, -8, -22, -12, -24, -10, -26, -16, -28, 12, -30, -31, -18, -14, -22, -22, -36, -16, -22, -26, -40, 12, -42, -28, -30, -20, -46, -28, -48, -22, -30, -34, -52, -24

Offset

1,3

Comments

The values of n which generate negative elements of this sequence are in A129487, the values of n which generate the zeros of this sequence are in A002827 and the values of n which generate positive elements of this sequence are in A034683

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Unitary Divisor.

Formula

a(n) = A034460(n) - n = A034448(n) - 2n.

From Amiram Eldar, Apr 06 2024: (Start)

a(A129487(n)) < 0.

a(A002827(n)) = 0.

a(A034683(n)) > 0.

Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(2*zeta(3)) - 1 = -0.3157836111... . (End)

Example

As the unitary divisors of 12 are 1, 3, 4 and 12, which sum to 20, then a(12) = 20 - 2*12 = -4.

Maple

A129468 := proc(n)
    A034448(n)-2*n ;
end proc:
seq(A129468(n), n=1..40) ; # R. J. Mathar, Nov 10 2014

Mathematica

UnitaryDivisors[n_Integer?Positive] := Select[Divisors[n], GCD[ #, n/# ] == 1&]; sstar[n_] := Plus@@UnitaryDivisors[n] - n; sstar[ # ] - # &/@ Range[40]
a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - 2*n; a[1] = -1; Array[a, 100] (* Amiram Eldar, Apr 06 2024 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, 1 + f[i, 1]^f[i, 2]) - 2*n; } \\ Amiram Eldar, Apr 06 2024

Crossrefs

Cf. A034460, A034448, A129487, A002827, A034683.

Cf. A002117, A013661.

Sequence in context: A065332 A265515 A336564 * A236846 A236847 A091703

Adjacent sequences: A129465 A129466 A129467 * A129469 A129470 A129471

Keyword

easy,sign

Author

Ant King, Apr 17 2007

Status

approved