A129468 - OEIS

%I #18 Apr 06 2024 07:57:21

%S -1,-1,-2,-3,-4,0,-6,-7,-8,-2,-10,-4,-12,-4,-6,-15,-16,-6,-18,-10,-10,

%T -8,-22,-12,-24,-10,-26,-16,-28,12,-30,-31,-18,-14,-22,-22,-36,-16,

%U -22,-26,-40,12,-42,-28,-30,-20,-46,-28,-48,-22,-30,-34,-52,-24

%N Unitary abundance of n.

%C 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

%H Nathaniel Johnston, <a href="/A129468/b129468.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.

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

%F From _Amiram Eldar_, Apr 06 2024: (Start)

%F a(A129487(n)) < 0.

%F a(A002827(n)) = 0.

%F a(A034683(n)) > 0.

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

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

%p A129468 := proc(n)

%p     A034448(n)-2*n ;

%p end proc:

%p seq(A129468(n),n=1..40) ; # _R. J. Mathar_, Nov 10 2014

%t UnitaryDivisors[n_Integer?Positive] := Select[Divisors[n], GCD[ #,n/# ] == 1&]; sstar[n_] := Plus@@UnitaryDivisors[n] - n; sstar[ # ] - # &/@ Range[40]

%t a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - 2*n; a[1] = -1; Array[a, 100] (* _Amiram Eldar_, Apr 06 2024 *)

%o (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

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

%Y Cf. A002117, A013661.

%K easy,sign

%O 1,3

%A _Ant King_, Apr 17 2007