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%I #19 Apr 25 2024 05:18:37
%S 0,-1,1,-2,5,2,10,-4,2,13,24,1,32,28,34,-9,52,5,62,21,64,67,86,-2,24,
%T 90,6,48,126,92,140,-18,141,144,159,-3,187,174,187,36,220,176,237,122,
%U 96,239,274,-7,72,65,292,167,331,12,335,89,350,351,393,160,414,392,184,-37,440,392,480,270,477,441,526
%N a(n) = floor(n*sqrt(gamma(n))) - sigma(n), where sigma(n) = A000203(n) is the sum of the divisors of n and gamma(n) = A007947(n) is the greatest squarefree divisor of n.
%C It appears that almost always sigma(n) <= n*sqrt(gamma(n)) (see A358308).
%D József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 77, section III.1.1.d.
%H Amiram Eldar, <a href="/A358309/b358309.txt">Table of n, a(n) for n = 1..10000</a>
%t a[n_] := Module[{f = FactorInteger[n], p, e}, {p, e} = Transpose[f]; Floor[n * Sqrt[Times @@ p]]- Times @@ ((p^(e+1)-1)/(p-1))]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Apr 25 2024 *)
%o (PARI) a(n) = {my(f = factor(n)); floor(n*sqrt(vecprod(f[, 1]))) - sigma(f);} \\ _Amiram Eldar_, Apr 25 2024
%Y Cf. A000203, A007947, A358308.
%K sign
%O 1,4
%A _N. J. A. Sloane_, Dec 09 2022
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